4. WAVES

4.1. Waves in one dimension (sections 4.1 - 4.6)

Oscillation

An oscillation is a back-and-forwards-movement like a mass hanging on a spring which is extended and released.

[In this case, when the force trying to make the oscillating object return to its equilibrium position follows the formula F = (-) kx like the force from a spring, the motion is called simple harmonic motion. The minus sign means that the spring is pulling or pushing with a force in the opposite direction to the displacement x. Since we have

F = -kx = ma we get x = (-k/m)a

that is, the displacement is some negative constant multiplied with the acceleration. If we look for a function to describe where an object is as a function of time, we can no longer use x = s = vt as for UM or x = s = ut + ½at² as for UAM since the force and therefore a is changing. If velocity describes how the displacement changes with time and acceleration how velocity changes with time, then these functions (plotted for x-values from 0 to 360^o or 0 to 2p) fit the bill:

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More specifically, velocity is the derivative of displacement and acceleration the derivative of velocity. You will learn in maths that the derivative of sin x is cos x and that of cos x is -sin x. For these reasons a sine function describes the wavelike motion we get when something is moving back and forward like a mass on a spring. A suitable function will be x(t) = A sin (2pft + P), the meaning of which is explained later].

Wave pulse

If the people at a football stadium "do the wave", there are two kinds of motion:

·        the back-and-forward motion of the hands

·        the motion of the "wave" travelling along the seats. This may be a uniform motion

If only one "wave" is sent out, it is a wave pulse.

Continuous wave

If wave pulses are sent out at a constant rate - like a hand setting a rope in motion with a series of wave pulses - it is a continuous wave.

Every wave pulse and continuous wave transfer energy (in the form of kinetic energy of the oscillating objects or particles, or in other forms)

Medium

Medium is the "material" which the wave (mostly) has to move through.

Examples of waves

Wave type Medium Oscillating "particle"

"the wave" people hands

ocean wave water water molecule

sound air (or other) air molecule (or other)

light does not need one, electromagnetic fields (later)

can move in vacuum

Transverse wave

These are waves where the oscillation is at a 90^o angle to the direction where the wave is moving. Ex. "the wave", ocean waves, light

Longitudinal wave

These are waves where the oscillation is parallel to the direction where the wave is moving. Ex. sound.

Graphs of waves - horizontal axis:

Here we can use either the time t which has passed since the first wave pulse we study was sent, or the distance or displacement s which the wave has traveled. If the velocity v of the wave is constant then s = vt and the shape of the wave is the same in either case.

Ex. if v = 10 ms^-1 and we have t = 1, 2, 3, ...seconds on the horizontal axis, then the graph with the displacement on the horizontal axis will look the same but have s = 10, 20, 30, ... meters there.

Graphs of wave - vertical axis:

On the vertical axis we place the displacement of the oscillating particle from its equilibrium position (that means, where it would be if there was no wave motion). Note:

·        this displacement may be much smaller than the displacement moved by the wave. If we send a sound across the room, the displ. traveled by the wave is several meters, but every oscillating air molecule maybe moves only small fractions of a millimeter back and forward.

· in a graph, we must have the axes at a 90^o angle to each other to see any curve. This makes the graph LOOK more like a transverse wave than a longitudinal - but it can be USED to illustrate both types of waves!

In the graph on the left, we have a plot of the displacement for many oscillating particles at different distances from a starting point but at one point in time (like a still photograph). 

In the graph on the right, we have the displacement of one oscillator graphed for many points in time, like if we had followed one particle with a video camera, frozen the film at many time points and graphed the displacement observed.

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Wave quantities - displacement on horizontal axis

Crest = the highest point on the wave graph

Trough = the lowest point

Equilibrium position = the horizontal axis, where the oscillator is if we have no wave

Wavelength l (lambda) = the distance between one crest and the following, or one trough and the following

Amplitude A = maximum displacement of the oscillating particle

Wave quantities -time on horizontal axis

Other quantities the same, but instead of wavelength we have:

Time period T = the time between one crest and the following or one trough and the following

from which we can define the number of full wave motions (with a crest, a trough, and two places where the graph is at the horizontal axis) :

the frequency f in the unit 1 hertz = 1 Hz = 1 s^-1

f = 1/T [DB p. 6]

Quantities specific to longitudinal motion

"Crest" and "trough" in the graph of a wave motion can in principle be used about both transverse and longitudinal waves, but somtimes we use for the longitudinal:

Compression = a place where the oscillating particles are closer than they otherwise would be

Rarefaction = a place where they are further apart

Wavelength and time period can for these waves be found using them instead of crest or troughs.

Wave speed (or velocity) v

The speed of the wave is the distance it travels by time (or the velocity the displacement by time). For one full wave motion, we have

distance = l time = T => speed v = l/T but since f = 1/T

this can be written:

v = f l [DB p. 6]

For a wave with a certain speed, this means that the higher the frequency, the lower the wavelength, and vice versa.

·        For sound (speed in air ca 340 ms^-1), the frequency or wavelength describes how "high" a tone is.

·        For light (speed in vacuum or air ca 300 000 000 ms^-1), they describe the color. (short l and high f for blue light, longer l and lower f for red light). Other colours in between.

[It can be shown that the displacement y as a function of time for the oscillating particle is y(t) = A sin (2pft + P), where the difference in travelled distance compared to another wave or a chosen point = the phase shift P = 2px/l

That means that for every wavelength l we move in the direction where the wave traveled, we add 2p to what we take the sine of, which gives the same result as if we had not added anything]

Electromagnetic waves

Of the mentioned wave types, electromagnetic waves are exceptional in that the oscillator is not a particle but electromagnetic fields, which will be explained later (although they to some extent can be interpreted as particles - even more about that even later). Common to them is the constant speed c = 300 000 000 ms^-1 in vacuum (and air). With v = c = lf => f = c / l it means that we have a high frequency when the wavelength is short and lower frequency when the wavelength is longer.

Type of EM - wave Wavelength l (m) Frequency f (Hz)

·        Gamma rays (one type 10^-13...10^-10 ca 10²⁰

produced in radioact.)

·        X-rays 10^-11...10^-8 ca 10¹⁸

·        Ultraviolet (UV) light 10^-9...10^-7 ca 10¹⁶

·        Visible light 10^-7...10^-6 ca 10¹⁵

violet (380..450 nm)

blue (450..490 nm)

green (490..560 nm)

yellow (560..590 nm)

orange (590..630 nm)

red (630..760 nm)

·        Infrared (IR) or heat 10^-6...10^-4 ca 10¹³

radiation

·        Microwaves 10^-4..10^-2 ca 10¹¹

·        TV, radio waves 10^-2..10³ 10⁴..10⁹

4.2. Superposition and interference in one dimension

If two waves are travelling in the same medium (here, we only study it along one straight line) then both waves are trying to affect the position of the oscillating particle.

Principle of superposition : the displacements caused by the two (or more) waves can be added (with their positive or negative signs)

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[Mathematically, we can add the displacements y₁ = A₁sin (2pf₁t + P₁) and y₂ = A₂sin (2pf₂t + P₂) and since we can choose where we put the origin of our coordinate system always have for example P₁ = 0)]

Constructive interference

If two waves have the same l (or the same f) and the phase shift is 0, 2p, 4p, 6p, ... they are strengthening each other and produce a resultant wave with a larger amplitude. (Exercise: Draw the graph of a wave and then the same phase-shifted 2p, plus the resultant wave.)

Destructive interference

If two waves have the same l (or the same f) and the phase shift is p, 3p, 5p, ... they are weakening each other and produce a resultant wave with a smaller amplitude. If A₁ = A₂ they may completely extinguish each other. (Exercise: Draw the graph of a wave and then the same phase-shifted p, plus the resultant wave.)

4.3. The phenomenon of "beats"

Ordinary interference is caused by two waves with the same frequency and wavelength. But what if the two waves have slightly different frequencies - like the sound of two tuning forks of which one is equipped with a clamp which slightly alters the frequency?

This can be simulated by making a spreadsheet produce a graph of the sum wave of two waves with the slightly different f₁ anf f₂. It will show a graph where the amplitude of the wave is periodically increasing and decreasing (although the amplitudes A₁ and A₂ are constant!). It will also be noted that the shape of the graph is not affected by a possible phase shift.

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The "beat frequency" which means how many times per second the amplitude of the sum wave is oscillating is:

f_beat = ½f₁ - f₂ ½ [DB p. 6]

4.4. Reflection in one dimension

Fixed end of rope

If you send a wave pulse along a rope fixed at one end, the pulse will be reflected because the oscillating particle at the end of the rope acts on the object it is attached to which then acts back on the particle with a force in the opposite direction (Newton's III law !) sending an inverted wave pulse in the opposite direction.

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Loose end of rope

If the end of the rope is left loose, a wave pulse reaching the end of the rope will find no more "rope particles" which could take the energy of the oscillation; the particles at the end of the rope will then be oscillating in the same direction as before but to a greater extent; which can be interpreted as a new pulse being started and sent in the opposite direction (but not inverted).

Other reflections

Other waves will also be reflected when they reach the end of the medium (if any) where they can travel. Light is reflected in mirrors but also from other surfaces, sound to some extent from solid surfaces.

4.5. Standing (stationary) waves

The guitar string: standing wave

·        If the string of a guitar is plucked, a wave pulse will be sent to the end where the string is attached (and also to the other end).

·        This wave pulse will be reflected and meet the reflected pulse from the other end (for instruments like the violin, where the string can be affected continuously, they may also meet new wave pulses being sent).

·        These reflected waves will be interfering with each other - constructively or destructively.

·        If the interference is constructive, the string may oscillate up and down at certain places which are not moving - the crests and troughs are switching place, but not moving along the string.

·        Although this standing (or stationary) wave is not moving, the waves which it is a sum are moving back and forward on the string.

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The places on the standing wave where the string is NOt Displaced are nodes (N)

The places where there is maximum displacement are called antinodes (A)

There are several (in principle, infinitely many) possible ways to have constructive interference: the fundamental or first harmonic, the second harmonic, the third etc.

Conditions for resonance giving a stationary wave: string fixed at both ends

"Resonance" of the waves on the string means that they interfere constructively. Examples:

Fundamental (first harmonic):

·        there must be nodes at the ends where the string is attached

·        between them, there must be one antinode

·        this only makes half the full traveling wave motion so if the length of the string is L we get

·        L = l/2 which is combined with v = fl => l = v / f gives :

·        L = (v/f)/2 = v/2f => f = v/2L = 0.5(v/L) = f₁

Second harmonic

·        now we have one full wave of the traveling wave motion in the string, so

·        L = l which with l = v / f gives

·        L = v / f and then f = v/L = 1.0(v/L) = f₂ = 2f₁

Third harmonic

·        now we have one full wave and half of the next in the string, so

·        L = 1.5l or L = 3l/2 which with l = v / f gives

·        L = 3(v/f)/2 = 3v/2f => f = 3v/2L = 1.5(v/L) =f₃ = 3f₁

This can be summed up in the formula:

f_n = n(v/2L) = nf₁, n = 1,2,3, ... [not in DB]

NOTE: The difference between f_n and f_n+1 is the same as f₁.

Conditions for resonance giving a stationary wave: pipe open at both ends

Sound can also be produced in the vibrating pillar of air in a tube-shaped instrument. Here the oscillations are longitudinal - parallel to the tube, but they can be illustrated with a graph showing the displacement of the air molecules from their ordinary (equilibrium) position as a function of the place in the pipe:

[Imagine an x-axis along the middle of the tube: these will then be the graphs of the displacement of the oscillating air molecules. The actual oscillation takes place parallel to the tube since sound is a longitudinal wave, although it must be graphed as if it were transverse].

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Fundamental (first harmonic)

·        now we must have antinodes (A) at the ends where the air molecules can oscillate freely and one node (N) in the middle

·        for the fundamental, we again have half a full traveling wave in the pipe length L (from crest to trough or trough to crest)

·        everything is mathematically the same as for the string fixed at both ends

Second harmonic

·        again, we have one full wave in the pipe now (from crest to crest)

Third harmonic

·        again, we have 1½ full travelling wave in the pipe (from crest to crest to the following trough or from trough to trough to the following crest)

This can be summed up in the formula - all same as for the string fixed at both ends:

f_n = n(v/2L) = nf₁, n = 1,2,3, ... [not in DB]

NOTE, again : The difference between f_n and f_n+1 is the same as f₁.

Conditions for resonance giving a stationary wave: pipe open at one end, closed at the other

Now the situation will be different.

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Fundamental (first harmonic)

·        we must have an antinode (A) at the open end where the air molecules can oscillate freely, but we have a node (N) at the closed end where the wall is stopping their oscillations (in a direction parallel to the pipe!).

·        this means that in the pipe length L we only have one fourth of a full traveling wave (from one place where there is no displacement to the next crest or trough) so

·        L = l/4 which with v = fl => l = v / f gives

·        L = (v/f)/4 = v/4f giving

· f = v/4L = 0.25 (v/L) = f₁

Third harmonic :

·        now we have 3/4 of a full travelling wave in the pipe (from no displacement to no displacement to the next crest or trough) so

·        L =3l/4 which with l = v / f gives

·        L = 3(v/f)/4 = 3v/4f giving

· f = 3v/4L = 0.75 (v/L) = f₂ = 3f₁

Fifth harmonic :

·        now we have 1.25 full travelling wave in the pipe (from one place of no displacement to the next = half a wave; then to the next = a whole wave, and on to the next crest or trough) so:

·        L = 5l/4 which with l = v / f gives

·        L = 5(v/f)/4 = 5v/4f and

· f = 5v/4L = 1.25 (v/L) = f₃ = 5f₁

That we get the frequencies f₁, 3f₁, 5f₁, ... explains we call them the first, third, fifth, ... harmonic. It can be summed up as:

f_n = n(v/4L) = nf₁, n = 1, 3, 5, ... [not in DB]

NOTE: The difference between f_n and the following frequency f_n+2 is the same as 2f₁.

4.6. The Doppler effect for sound

The ambulance passing by ... and passing a sound signal on a train

If an ambulance is approaching, the sound of its sirens is higher than if it was standing. When it is moving away, the sound is lower, and when it passes us, the sound frequency changes clearly.

If we sit on a train and it passes a railroad crossing with a sound signal, this sound is higher than normal when we approach it and lower when we have passed it and are moving away.

Moving source, stationary observer (ambulance)

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·        The source sends out sound with the sound speed v of frequency f.

·        If nothing moves, the distance between crests = l

·        But if the source is approaching us with the speed v_s, it will have moved the distance vt towards us in the time it took to send out one full wave; that is the time period T.

·        so the distance between the crests is actually l - v_sT which is the new wavelength l'.

·        the speed of sound is the same so v = lfbut also v = l'f' which gives f' = v/l'

·        then we get f' = v/ (l - v_sT)

·        on the left hand side, we can divide with something both "upstairs" and "downstairs" like when 2x = 6/8 gives 2x = 3/4 if both 6 and 8 are divided with 2

·        what we divide with is T giving v/T = vf upstairs

·        in the parenthesis downstairs both terms must be divided with T; the first gives l/T = lf = v

·        the second gives v_sT/T = v_s

·        our equation is now f' = vf / (v - v_s)

·        if we now on the left hand side divide with v both upstairs and in both terms downstairs we get f' = f / (1 - v_s/v) which can be written

· f' = f ( 1 / ( 1 - v_s/v) )

If the source instead had been moving away, the new l' = l + v_sT and only the sign in the parenthesis would have been different.

Moving source: f' = f ( 1 / (1 ± v_s/v)) [DB p.6]

where the positive sign is for a source moving away, the negative for one approaching.

Moving observer, stationary source (sitting on a train and passing a sound signal)

Now there is no new wavelength, since the source is not moving and the crests therefore sent out with the same distance between each other. But since the observer is moving towards the wave with the speed v_o, the relative speed is added (like if you collide head on with something). So:

·        the new relative speed v' = v + v_o but v' = f'l' = f'l so

·        v + v_o = f'l or f'l = v + v_o but since v = fl gives l = v/f we get

· f'(v/f) = v + v_o where both the left and right hand side is multiplied with f/v so

· f' = (f/v)( v + v_o) which can be written as:

· f' = f ( v / (v + v_o) ) and in the parenthesis the upstairs part and both terms in the downstairs part are divided with v giving

· f' = f (1 + v_o/v)

If the observer had instead been moving away from the source, everything would be the same except that the relative speed would have been v' = v_o - v (like something colliding from behind) and the sign in the parenthesis negative.

Moving observer: f' = f (1 ± v_o/v) [DB p.6]

The formulas are in the data booklet, but how do I remember the signs? If they are getting closer, the f' should be higher than f, if they are getting further away, f' is lower than f. Use a sign that makes this happen.

[If both the source sending the f and the observer are moving, use the source formula to find a theoretical frequency f_t which in reality is never sent or heard, and then apply the observer formula to that f_t to find the f' which is heard :

f_t = f ( 1 / ( 1 - v_s/v) ) and then f' = f_t (1 + v_o/v) = f ( 1 / ( 1 - v_s/v) ) (1 + v_o/v) if both are moving towards each other; other signs in other situations]

The Doppler effect for other (EM) waves

A similar Doppler effect can be observed for other types of waves than sound, although the formula for it and the proof of it are somewhat different.

·        For radar (a type of radio) waves they can be reflected back from a car and the reflected wave has a different frequency which depends on how fast the car moves

·        Light from distant stars is shifted towards a lower frequency or higher wavelength (towards the red colour) if it is (as is often the case) moving away from is ("redshift")

Reflection, Doppler and beats

If a wave is sent and reflected from a moving object giving it a slightly different frequency the original wave and the reflected, Doppler-shifted wave may produce a beat phenomenon; that is an oscillation in amplitude at a frequency equal to the difference between their frequencies.

4.7. Waves in two dimensions (section 4.7. - 4.10.) : Huygen's principle & diffraction

Rings on the water

So far we have been working with waves moving in one dimension - along a rope or string, in a long pipe, or similar. But in reality waves can move in 2 or even 3 dimensions. We will first focus on waves in 2 dimensions - for example the waves on the surface of a lake or pond when a stone is dropped into it.

Wave fronts

The rings which are formed are places where the water molecules have a maximum displacement upwards, and are formed by the crests of the waves moving from the place where the stone was dropped out in all directions.

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Rays

A line showing in what direction the 2-dimensional wave is moving is called a ray or beam.

Wavefronts are at a 90^o angle to the rays

For waves on a water surface, the wavefronts (= crests) are easy to see; for a narrow beam of light we see the ray.

Huygen's principle

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 From every point (every water molcecule) on a wavefront (here a straight one) new small wavefronts ('wavelets') are sent out in all directions except "backwards" where the previous wavefront in coming. They are shaped like half-circles, but together they form a new wavefront of the same shape as the previous one.

Diffraction

If a wavefront has to pass a narrow gate or opening (aperture), the new wavefront may not be of exactly the same shape as the previous.

·        If the opening and the wavelength l are about of the same magnitude, much of the wave will spread in half-circle shaped new wavefronts after the gate.

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Example : Sound moves at 340 ms^-1. A typical sound has the frequency f = 440 Hz. From this the is obtains as v = fl => l = v / f = around one meter. This is also the size of a typical door. So sound diffracts well at a door and therefore we can hear sounds "around the corner".

·        If the opening and the l are of very different magnitudes, there will be little diffraction.

Example : visible light has a l = about 400 to 800 nm (nanometers), much smaller than the door. Therefore we can not see around a corner. To diffract light, we need a much smaller opening.

Note : Diffraction may also happen when a wave reaches an obstacle - for example waves on the surface of water hitting a tree or plant. Then new waves are sent out in all directions from this obstacle. A log floating in the water diffracts the waves if its length if of the same magnitude as the l of the waves.

Exercise: Draw the wavefronts when planar water waves hit a) a point obstacle, like a reed or pole in the water b) an obstacle of the same size as the wavelength, like a log floating in the water

4.8. Superposition and interference in two dimensions - Young's experiment

Superposition and interference - review

Recall that if two waves are affecting the same oscillating particle (or other oscillator) the resultant wave will have the sum of the displacements of them. If the phase shift between them is 0, l, 2l, 3l , ... the interference will be constructive and the resultant wave strengthened.

Diffraction in two openings

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·        If a wavefront has to pass two openings with a width similar to the l, these openings will act as new sources (S₁ and S₂) of half-circle shaped new wavefronts

·        let the distance between the centers of the openings be d and the distance to place where we can study what has happened D (if the wave is a light wave, this may be a screen or a white paper)

·        we study a point on this screen such that the angle between the original direction of the wave and the direction from S₁ or S₂ or a point in between them is q. If d < < D (d is much smaller than D) it makes little difference which we take.

·        the phase shift is the distance from S₂ to X, which we call S₂X

·        from the geometry of the situation it follows that S₂X/d = sin q.

·        we have constructive interference if S₂X = 0, l, 2l, 3l, ...nl

·        so we get bright spots on the paper if nl/d = sin q or

d sin q = n l [DB p. 6]

where n = 0, 1, 2, .... (called zeroth, first, second, ... order maxima)

Note:

·        we get the n = 1, 2, 3, .. maxima symmetrically on both sides of the n = 0 maximum

·        if we have more than two slits all with the same distance d between the centers of the openings, the phase difference will be a whole number of l for them all and the same formula will apply. Now the situation is much more sensitive to a change in q and therefore only a very precise value of this angle will give constructive interference => the bright spot is small

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·        then number of bright spots is limited since sin q cannot be more than one, and solving for n gives n = d sin q/ l so the maximum n is the whole number equal to or less than d / l

Another way of expressing the same formula is to use the distance s between the bright spots on the screen and the distance D from the grating to the screen:

• using the angle q between the rays to the n = 0 and n = i maxima we have sin q = is/D
• inserting this into d sin q = nl gives d(is/D) = il, cancelling i gives
• ds/D = l which solved for s becomes:

s = lD / d [DB p. 6]

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4.9. Reflection in two dimensions

Reflection at a plane surface

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As is explained below this picture, it can be shown that at a plane surface, the

angle of incidence = angle of reflection

where both angles are measured from the normal to the surface (it is here also true that the angles to the surface are the same).

The same can be illustrated using either wavefronts or rays, and applies not only to light but to any 2-dimensional wave which reaches a surface it cannot penetrate:

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Reflection at a curved surface

At a curved surface, we can select a few points at the surface, draw the normal to the surface at these points, and construct where the rays will be going after the reflection. The wavefronts are as usual perpendicular to them.

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4.10. Refraction

Refraction = bending

Refraction is the phenomen that light rays (and other waves) bend (are "broken") when they pass the boundary surface between two media - for example air and water. This leads to the phenomenon that an oar sticking down in the water appears to be bent. Allt this can also be explained using Huygen's principle for a plane wavefront reaching the boundary between the media 1 and 2.

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The angle of incidence q₁ or i and angle of refraction q₂ or r are measured from the normal to the boundary surface.

Snell's law

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In this picture, the original wavefront is the line AX and from it wavelets are spreading in half-circles.

·        from the point A the half-circle has the radius AY

·        from the point X the half-circle has the radius XB

A new wavefront is formed along the line BY. From the geometry and trigonometry of the siutation we get, using the symbols v₁ and v₂ instead of c₁ and c₂ for the speeds of the wave in the media 1 and 2 (Snell's law):

sin Q₁ / sin Q₂ = v₁ / v₂ [DB p. 6]

Refractive index

Since the speed of light in vacuum and, approximately, in air is the constant c = 300 000 000 ms^-1 it may be convenient to define the refractive index n for a material using v = the speed of light in the material as :

n = c / v [DB p. 6]

For light traveling between two materials 1 and 2 we then have:

·        n₁ = c/v₁ => v₁ = c/n₁ and n₂ = c/v₂ => v₂ = c/n₂

·        inserting in Snells law gives:

·        sin Q₁ / sin Q₂ = (c/n₁) / (c/n₂) which cancelling c gives

·        sin Q₁ / sin Q₂ = (1/n₁) / (1/n₂) which is

·        sin Q₁ / sin Q₂ = n₂ / n₁ and crosswise multiplication gives us :

n₁ sin Q₁ = n₂ sin Q₂ [DB p. 6]

Refractive indices for som materials:

Air 1.0003 (use 1 unless otherwise stated)

Water 1.33

Glass 1.5 to 1.6

Diamond 2.4