# Sound Waves

## Travelling Longitudinal Waves

In a longitudinal wave, the wave elements oscillate parallel to the direction of travel. This causes the elements to compress together and then stretch out as the wave passes.

### Pressure Variations

 When longitudinal waves pass through a gas, they cause variations in pressure as the gas particles compress/separate. High pressure regions are known as compressions, while low pressure regions are rarefactions.

For example, imagine a speaker diaphragm which is pushed to the right. As it moves, it pushes the gas particles in front of it, compressing them. The diaphragm stops, but the high pressure region (compression) continues travelling through the gas. When the diaphragm moves back again, it creates a low pressure region (rarefaction) which also propagates through the gas in the same direction as the high pressure region. These pressure regions can each be thought of as a longitudinal equivalent to the transverse pulse on a string. If the diaphragm now moves back and forth with simple harmonic motion, it produces a regular series of compressions and rarefactions, which propagate through the gas. This is a longitudinal wave. Each individual gas particle is now oscillating (parallel to the wave) with simple harmonic motion.

The wavelength of a longitudinal waves is the distance between two consecutive compressions (or two consecutive rarefactions). The displacement of an individual particle (of initial location x) in a longitudinal wave is given by the following equation:

s(x,t) = smax cos(kx-ωt) , smax is the displacement amplitude.

A more useful way of looking at longitudinal waves through gases is by examining pressure. The following equation gives change in pressure at location x in the wave:

ΔP = ΔPmax sin(kx-ωt) , ΔPmax is the pressure displacement.

Note that ΔP measures a pressure difference, i.e. a pressure above or below the original pressure. It is also important to see that the displacement equation uses cosine, while the pressure equation uses sine - the two are not maximum at the same time. In fact, displacement is 0 at compressions and rarefactions, while ΔP is 0 at displacement maxima/minima.

### The Speed of Sound

The speed of all waves depends on mechanical and inertial properties of the wave:

v = sqrt(elastic property/inertial property)

In solids, the density and Young's modulus (elasticity) define the speed of sound.

vsolid = sqrt(Y/ρ)

In liquids, the density and bulk modulus define the speed of sound.

vliquid = sqrt(B/ρ)

Gases are a little more complicated, since temperature also becomes important. In air, the relationship between temperature (in Celsius) and speed of sound is given by the following equation:

'vair = sqrt331(1 + (T/273))

Remember that sound is a mechanical wave, and so cannot travel in a vacuum.

### Wave Fronts and Rays

• A wave front is a surface where the phase of a wave is constant.
• Wave fronts are perpndicular to the wave's direction of travel.
• The distance between two wave fronts with the same phase is equal to the wavelength.
• "In phase" is another way of saying "at the same point in the oscillation cycle". The wave elements on a wave front are in phase, and so will all have the same displacement and motion. In a sound wave, the pressure (and particle displacement) are constant along the wave front. The graphs of two waves which are in phase will "line up".
• Rays are lines pointing parallel to the wave direction, from the wave source to the wave front.

### Energy, Power and Intensity of Sound Waves

 Sound waves are comprised of kinetic and potential energy. The kinetic energy may be found using k=½mv2 and the speed of an individual particle (time derivative of the displacement equation above). This gives k = ¼ρA(ωsmax)2λ. The total potential energy is the same as total kinetic energy, which means that the total wave energy is given by:

E = K+U = ½ρA(ωsmax)2λ

Power is the rate of energy transfer of the wave. It is the amount of energy which passes a fixed point during one oscillation.

Power = ΔE/Δt = ½ρAv(ωsmax)2

Intensity is power per unit area. The area A describes the area of the wave front. The wave power is spread uniformly across the wave front.

I = Power/A = ½ρv(ωsmax)2 = (ΔP2)/(2ρv)

Often, sound waves originate from a point source, which emits waves equally in all directions. This produces spherical waves expanding out from the source. The power of the wave is distributed evenly over the wave fronts, which each have a surface area of 4πr2. Using the intensity equation then produces the inverse square law:

I = Power/A = Power/(4πr2)

#### Decibels

Sound level is often measured using the logarithmic decibel scale, since the human ear can detect an extremeley wide range of intensities. 120dB is the threshold of pain, and continued sound above 90dB is damaging to the ear. 0dB is the threshold of hearing (in decibels).

β = 10log(I/I0)

β is the sound level in decibels. I is the intensity, and I0 is the reference intensity, taken at the threshold of hearing. I0 = 1*10-12 W/m2.

## The Doppler Effect

 The Doppler effect causes a change in the frequency of waves, depending on the motion of the observer and the source. When the source is moving towards the observer, the frequency is higher, and vice versa.

Case 1 - Stationary Observer, Stationary Source: A stationary point source emits waves with a fixed wavelength (λ) travelling at speed (v) in all directions. The frequency at the source is f=v/λ. The stationary observer receives the waves with the same wavelength, speed and frequency. There is no Doppler effect.

Case 2 - Moving Observer, Stationary Source: The source is emitting waves as before, but the observer is now moving towards the source with velocity (vo). The observer still receives the waves at the same wavelength (λ), but his movement relative to the source means he sees the waves as moving faster than they actually are. The observed frequency is thus f' = v'/λ = (v+vo)/λ. Since λ=v/f, we can write f' = ((v+vo)/v)f. If the observer was moving away, the '+' in the numerator would be a '-'.

Case 3 - Stationary Observer, Moving Source: The source is emitting waves as before, but is now moving towards the stationary observer with speed (vs). The observer receives the waves at speed v (NOT v+vs), since the wave fronts travel at the same speed regardless of the wave source speed. However, the distance between wave fronts is smaller than before, since the source is moving towards the wave fronts it previously emitted. The observed frequency is thus f' = v/λ' = (v/(v-vs))f. If the source was moving away, the '-' in the denominator would be a '+'.

Combining the equations for cases 2 and 3 gives a general equation for when the source and observer are both moving towards each other. If one of them is moving away, flip the relevant sign as described as above.

f' = ((v+vo)/(vsvs))f

Note that the doppler effect acts along the line of motion between the object and observer. If any object of not moving directly towards (or away from) an observer, consider only the component of motion towards (or away from) the observer.

If a sound source is moving faster than the speed of sound in that medium, is will create a conical wave front (with the point at the sound source). The apex half-angle of this cone is calculated as θ = sin-1(v/vsound). θ is the Mach Angle, and the Mach number is vsound/v. The conical wave front carries a lot of energy, and causes a considerable pressure variation when it passes.

## Standing Longitudinal Waves

Standing waves can be set up in air columns (in pipes), when sound waves travelling in opposite directions interfere with one another. These standing waves behave in a very similar manner to those on a string.

• A closed end of a pipe corresponds to a pressure antinode (displacement node). The air particles cannot oscillate longitudinally, forcing a displacement node.
• An open end of a pipe corresponds to a pressure node (displacement antinode). The particles can oscillate freely, but the pressure is forced to be equal with that outside the pipe.
• There are different modes of oscillation (harmonics) just as there are for standing waves on a string.
• If both ends of the pipe are open, all the harmonics can be played.
• If only one end is open, only the odd harmonics can be played. The pressure node at one end and antinode at the other mean that only the odd vibrational modes are possible.

### Beats

The interference discussed so far has been spatial interference, where the amplitude of oscillation depends on the location of an the oscillating element. The phenomenon known as beating arises from temporal interference.

• Temporal interference occurs then two waves are periodically in and out of phase.
• Temporal interference occurs when there is a slight difference between the frequencies of two waves.
• The two waves alternate between constructive and destructive interference over time. This causes a periodic variation in amplitude, known as beating.
• The number of amplitude maxima per second is the beat frequency. Humans can detect beat frequencies up to 20Hz.
• Beat frequency is equal to the difference in frequency between the two waves.

fbeat = |f1 - f2|

## End

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