Three sources of sound are distributed along a line, as shown in the figure. v is the speed of sound. Source A is at rest, and it emits sound waves with a frequency of 480 Hz. Source B is traveling at v/5 to the right, and it emits sound waves with a frequency of 960 Hz. Source C is traveling at a speed of v/3 to the left, and it emits sound waves with a frequency of 1550 Hz. A, B, and C also detect the sound waves emitted by the other sources. All the questions below pertain to when the three sources of sound are at the locations shown in the figure below.
a)What frequency does A observe for the waves emitted by B? (A = observer; B = source)
b) What frequency does C observe for the waves emitted by A? (C = observer; A = source)
c)What frequency does B observe for the waves emitted by C? (B = observer; C = source)
a. Fa = ((V-Va)/(V+Vb))*Fb
Fa = ((343-0)/(343+68.6))*960 = 800 Hz
b. Fc=((343+114.33)/(343-0))*480=640 Hz.
c. Fb = ((343+68.6)/(343-114.33))*1550
Fb = 2790 Hz.
To answer these questions, we need to consider the Doppler effect. The Doppler effect describes the change in frequency of a wave (in this case, sound waves) due to relative motion between the source and observer.
a) To find the frequency observed by A for the waves emitted by B, we need to consider that B is moving towards A. In this case, the formula for the observed frequency (f') is given by:
f' = f * (v + v_B) / (v + v_A)
where f is the emitted frequency, v is the speed of sound, v_B is the velocity of B, and v_A is the velocity of A.
In this case, A is at rest (v_A = 0) and B is traveling at v/5 to the right (v_B = v/5). Therefore, the observed frequency (f') for A will be:
f' = 960 * (v + v/5) / (v + 0)
Simplifying the expression, we get:
f' = 960 * (6v/5) / v
= 1152 Hz
Therefore, A observes a frequency of 1152 Hz for the waves emitted by B.
b) To find the frequency observed by C for the waves emitted by A, we again use the Doppler effect formula. In this case, C is moving towards A, so the observed frequency (f') is given by:
f' = f * (v + v_A) / (v + v_C)
Where f is the emitted frequency, v_A is the velocity of A, and v_C is the velocity of C.
In this case, C is traveling at v/3 to the left (v_C = -v/3). Therefore, the observed frequency (f') for C will be:
f' = 480 * (v + 0) / (v - v/3)
Simplifying the expression, we get:
f' = 480 * (3v/3) / (2v/3)
= 720 Hz
Therefore, C observes a frequency of 720 Hz for the waves emitted by A.
c) To find the frequency observed by B for the waves emitted by C, we once again use the Doppler effect formula. In this case, B is moving away from C. The observed frequency (f') is given by:
f' = f * (v + v_B) / (v + v_C)
Where f is the emitted frequency, v_B is the velocity of B, and v_C is the velocity of C.
In this case, B is traveling at v/5 to the right (v_B = v/5) and C is traveling at v/3 to the left (v_C = -v/3). Therefore, the observed frequency (f') for B will be:
f' = 1550 * (v + v/5) / (v - v/3)
Simplifying the expression, we get:
f' = 1550 * (6v/5) / (2v/3)
= 2790 Hz
Therefore, B observes a frequency of 2790 Hz for the waves emitted by C.