THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 438 Assignment # 6:
 
ACOUSTICS
 
SOLUTIONS:
 
Thu. 15 Mar. 2007 - finish by Thu. 29 Mar.

1. SHARK ATTACK: A diver makes a deep dive wearing a facemask that covers eyes, nose and mouth. At 35 m depth he suddenly notices a shark cruising towards him; he lets out a short shriek at the top of his voice with an intensity of 95 dB. What is the intensity of his voice transmitted into in the water just outside his face mask? (Neglect the acoustic impedance of the lens of his face mask; just consider the transition of sound from the air to the water).¹  ANSWER:  A diver at a depth of 35 m will encounter a pressure of 4.5 atm (1 extra atm per 10 m depth). That means the pressure is 4.5 times higher than at sea level. From the ideal gas law pV=nRT we see that (for constant temperature T and volume V) the molar concentration n/V=p/RT is proportional to the pressure p, and so is the density $\rho$. Since the pressure is increased by a factor of 4.5, the density is increased by a factor of 4.5 as well. Since the impedance Z is defined as $Z = \rho v$ (where v is the velocity of sound in air, which is approximately independent of pressure), this means that the impedance of air at 35 m depth is likewise increased by a factor of 4.5. From $I_{\rm trans} = 4.5 \times (Z_{\rm a}/Z_{\rm w}) \times I_{\rm a}$ we get $I_{\rm trans} = 7 \times 10^{-7}$ W/m² which corresponds to .

2. LONG DISTANCE TALK OF WHALES:
   1. Determine the critical angle of total internal reflection for sound waves in the Sofar channel. What is the beam angle $2 \delta$ (critical angle of TIR) into which an animal should emit its voice in order to match the Sofar channel sound wave guide?

      The Sofar channel.
      
      

   2. Suppose a whale in the Sofar channel talks to a friend 2000 km away. The animal emits a sound signal of power P = 1.2 W and frequency f = 10 Hz into a conical beam with the half angle $\delta$ calculated above. What is the intensity of this sound wave at a distance d = 2000 km? Express your result as a sound level $\beta$ [in dB] using $I_{\rm ref} = 10^{-12}$ W/m². With reference to Fig. 7.16 on p. 254 of the textbook, assume h = 800 m. Consider both the spreading of the wave with distance and the attenuation due to absorption.   ANSWER:  A full solution is on pp. 259-260 of the textbook.

   3. Determine the displacement amplitude s₀ and the pressure amplitude $\Delta p_0$ of the sound signal at the distance d = 2000 km.   ANSWER:  A full solution is on pp. 259-260 of the textbook.

   4. How long does it take for the message to travel from the sender to the receiver?   ANSWER:  A full solution is on pp. 259-260 of the textbook.

3. VOICES:
   1. Assume that elephants and mice roar like organ pipes, closed at one end. What is the lowest frequency of their voices? (Hint: You must guess or find from physiology texts the length of their "trumpets".)   ANSWER:  For a pipe of length L, closed at one end and open at the other, the $n^{\rm th}$ "mode" has lambda_n = 4L/(2n-1) and $f_n = v/\lambda$ = (2n-1)(356 m/s)/(4L), where we assume moist air at $37^\circ$C. The lowest frequency mode has n=1. The "organ pipe" of a typical human is taken to be about 17 cm, or about 20% of the length of the "trunk" of the body. Thus if the mouse's body (not counting legs or head) is about 5 cm then we would expect $L_{\rm mouse} \approx 0.01$ m, giving $f_1(\hbox{\rm mouse}) \approx 356/0.04$ or . A bigger mouse should have a lower voice. The elephant is another story, since its "trumpet" may or may not include the long trunk. If it does not include the trunk, then a 4 m long elephant may have $L_s \approx 0.8$ m, giving a "high low" of . If we assume the elephant is good at "talking through its nose" then we must add a trunk length of about 1.7 m to get $L_\ell \approx 2.5$ m, giving a "low low" of .²

   2. Bass singers produce sounds somewhat like Helmholtz resonators, where the lips and mouth from a pipe shaped opening of area A (a few cm²) and length L in which a plug of air resonates, driven by the flow of air from the lungs. Suppose a certain singer has a lung volume V = 7.5 liters and he opens his mouth to A = 1.5 cm². To what length L does he have to shape his "mouth pipe" in order to produce the frequency f = 65 Hz, and what tone is that?³  ANSWER:  Equation (9.41) says $f = (v/2\pi)\sqrt{A/LV}$. Solving for L gives $L = (v/2\pi f)^2 A/V$ = [(356 m/s $)/(2\pi \times 65$ Hz )]² x (1.5 x 10^-4 m ²)/(7.5 x 10^-3 m³) = 0.0152 m or .⁴The tone can be looked up: low-low C or C₂ [two octaves below "middle C"(C₄)] is listed as 65.41 Hz and the nearest other tones are B₁ at 61.74 Hz and C$_{\sharp2}$/D$_{\flat2}$ at 69.30 Hz, so the baritone is a little off-key at C₂.

   3. When frogs croak they blow up a part of their skin above an air sack. Approximate the skin (typically A = 10^-4 m², m = 10^-5 kg) as a piston driven by an air spring in a cylinder, and determine what sound frequency this device would produce. Compare this frequency to the frog data in Fig. 9.28 on p. 346 of the textbook, and comment.   ANSWER:  Here, since the effective oscillating mass is given, we use the equation (9.37) for the frequency for a piston with mass m: $f = (v/2\pi)\sqrt{\rho A/mL}$. Assuming a cylinder depth of L = 1 cm (i.e. similar to the diameter of the tympanum) we get a frequency of roughly which could be an E. portoricensis or possibly an E. coqui (see p. 346 of the textbook).

   4. Name three animals other than mammals, frogs or birds that produce sounds; at least one of these should be under water, and at least one on land.⁵  ANSWER:  Our good friend the mole cricket is joined by a chorus of other insects who make sounds by rubbing their legs together (evidently with tremendous enthusiasm!). In the water one finds shrimps making noise by cavitation as they snap their tails. Reef fish make quite a racket too, as they crunch bits of coral or barnacles with their teeth. (Anyone who has been snorkeling in the presence of either is familiar with these relatively boring sounds.) Whether these sounds are used for communication is an open question.