Acoustic Phenomena

(a) (10–20) Hz

Infrasonic waves have frequencies below 20 Hz, the lower limit of human hearing. Of the given ranges, only 10–20 Hz lies below 20 Hz.

(a) Doppler effect: The apparent change in the frequency (pitch) of a sound heard by an observer, when there is relative motion between the source and the observer, is called the Doppler effect.

(b) Observer moving towards a stationary source

Let the source emit frequency $f$, speed of sound $v$, so wavelength $\lambda = v/f$. The source is at rest, so the wavelength does not change. The observer moves towards the source with speed $u_o$.

1. Relative speed of sound with respect to the observer:

2. Number of waves received per second (apparent frequency):

3. Therefore

Since $f' > f$, the pitch heard is higher.

(c) Motion detector

Given: $f = 150$ kHz, truck speed $u = 120$ km/h $= \dfrac{120\times1000}{3600} = 33.3$ m s$^{-1}$, $v = 340$ m s$^{-1}$.

Step 1: The truck is a moving observer approaching a stationary source:

Step 2: The truck reflects $f_1$ and now acts as a moving source approaching the stationary detector:

Substitution:

Frequency of the reflected wave $\approx$ 183 kHz.

(b) $I \propto a^2$

The energy of a vibrating particle is proportional to the square of its amplitude, so the intensity of sound is $I = 2\pi^2 f^2 a^2 \rho v$, i.e. $I \propto a^2$.

Doppler effect is the apparent change in the frequency (pitch) of a wave heard by an observer when there is relative motion between the source and the observer. The pitch rises when they approach each other and falls when they move apart, for example the horn of a passing vehicle.

We can recognise a friend by his voice because of the quality (timbre) of sound.

When a person speaks, the vocal cords produce a fundamental note together with a number of overtones. The number of overtones and their relative intensities depend on the shape and size of each person's vocal cords, throat, mouth and nasal cavities. So two people speaking at the same pitch and loudness produce sounds of different waveforms. Our ear and brain identify this characteristic waveform, so we can tell who is speaking without seeing him.

i) Yes, the pitch changes. In figure (ii) the $f$ waves emitted in one second are spread over a length $V + V_s$ instead of $V$, so each wavelength is longer. This means the source is moving away from the listener. A longer wavelength reaching the listener at the same speed $V$ means a lower frequency, so the pitch heard is lower (it would be higher if the source moved towards the listener).

ii) Expression

Let the source emit frequency $f$, speed of sound $V$, source speed $V_s$ (moving away from L), listener at rest.

1. Source and listener at rest (figure i): $f$ waves occupy length $V$, so

2. Source moving away (figure ii): in one second the first wave travels $V$ towards L while the source moves $V_s$ away. The same $f$ waves now occupy length $V + V_s$:

3. The waves reach the listener at speed $V$, so the apparent frequency is

Since $V + V_s > V$, $f' < f$: the apparent pitch is lower, which supports the answer in (i).

(If the source moves towards the listener, $\lambda' = \dfrac{V - V_s}{f}$ and $f' = \dfrac{V}{V - V_s}f > f$, so the pitch rises.)

Given: $f = 400$ Hz, $v = 340$ m s$^{-1}$, change in frequency as the car passes $= 60$ Hz, car speed $= u$.

Formulae:

Change in frequency:

Substitution:

Velocity of the car $\approx$ 25.4 m/s (about 91 km/h).

(Since $u \ll v$, the approximation $\Delta f \approx \dfrac{2fu}{v}$ gives $u = \dfrac{60 \times 340}{2 \times 400} = 25.5$ m/s, nearly the same.)

Given: $f = 600$ Hz, train (and driver) speed $u = 10$ m s$^{-1}$, $v = 340$ m s$^{-1}$.

Step 1: Frequency reaching the cliff. The train is a source moving towards the stationary cliff:

Step 2: Frequency heard by the driver. The cliff reflects $f_1$ as a stationary source, and the driver is an observer moving towards it:

Substitution:

Frequency of the echo heard by the driver $\approx$ 636 Hz.

Given: $f = 500$ Hz, source speed $u = 20$ m s$^{-1}$, $v = 340$ m s$^{-1}$, observer at rest.

While approaching:

While receding:

Change in frequency:

The observer hears the frequency drop by about 59 Hz as the car passes.

(a) 0%

With source and listener moving in the same direction at the same speed $u = v/2$, the apparent frequency is

There is no relative motion between them, so no change in frequency (0%).

(a) Intensity from a point source follows the inverse square law, $I \propto \dfrac{1}{r^2}$. If the distance becomes four times, the intensity becomes $\dfrac{1}{4^2} = \dfrac{1}{16}$ of its original value. (If the distance is reduced to one quarter, the intensity becomes 16 times.)

(b) Two trains moving in opposite directions

Given: $f = 262$ Hz, source speed $u_s = 30$ m s$^{-1}$, observer speed $u_o = 18$ m s$^{-1}$, $v = 340$ m s$^{-1}$.

General formula:

(i) Approaching each other (observer moves towards source, source moves towards observer):

(ii) Receding from each other:

Frequencies heard: about 303 Hz while approaching and 228 Hz while receding.

(c) Maximum displacement

Given: pressure amplitude $P_0 = 3.0\times10^{-2}$ Pa, $f = 1000$ Hz, $v = 344$ m s$^{-1}$, bulk modulus $B = 1.42\times10^5$ Pa.

1. For a sound wave $y = s_0\sin(\omega t - kx)$, the excess pressure is $p = -B\dfrac{\partial y}{\partial x}$, so its amplitude is

2. Wave number:

3. Substitution:

Maximum displacement $\approx 1.2\times10^{-8}$ m, which is much smaller than the size of a dust particle.

(d) wave form

Quality (timbre) depends on the number and relative intensities of the overtones present with the fundamental, which together decide the shape of the waveform. Frequency decides pitch and intensity decides loudness.

The flute produces $f = \dfrac{v}{2L} = \dfrac{330}{2 \times 0.30} = 550$ Hz. For the passenger inside the bus it stays 550 Hz, because he moves with the source. For the person on the bench, the source is moving, so the Doppler effect applies.

• If the bus moves towards the person: the waves are crowded together and the apparent frequency is

so he hears a higher pitch.

• If the bus moves away from him: the waves are stretched and

so he hears a lower pitch.

As the bus moves uniformly, the pitch stays steady at the higher value while it approaches and drops suddenly to the lower value as it passes him. The faster the bus, the bigger the change.