Question

An airplane traveling at half the speed of sound emits a sound of frequency 5.29 kHz.

(a) At what frequency does a stationary listener hear the sound as the plane approaches? kHz

(b) At what frequency does a stationary listener hear the sound after the plane passes? kHz

Answer 1

Let speed of the sound be V_s = 340 m/s

Speed of the airplane V = V_s/2 = 340/2 =170 m/s

Frequency of the sound f = 5.29 kHz = 5.29 * 10³ Hz

Case 1) here airplane is moving towards the listener so the apparent frequency is

f¹ = f * ( V_s/V_s-V )

                    = 5.29 * 10³ * ( 340/340-170)

                    = 5.29 * 10³ * 340/170

                   = 5.29 * 10³ * 2 = 10.58 * 10³ Hz

Case 2) here the airplane is moving away from the listener so the apparent frequency is

f¹¹ = f * ( V_s/V_s + V )

                     = 5.29 * 10³ * ( 340/340 + 170 )

                     = 5.29 * 10³ * 340/510

                     = 5.29 * 10³ * 2/3

                     = 3.52 * 10³ Hz

So the apparent frequency of the sound heard by listener

1.when plane is approaching is f¹ = 10.58 * 10³ Hz = 5.29 kHz

2. when the plane is moving away is f¹¹ = 3.52 kHz