Question

Question: A cyclist ringing a bell with a frequency of 658 Hz rides towards a wall (Figure 1). The wall reflects the note back towards the cyclist. A sound meter detects a beat frequency of 13.0 Hz between the bell and the reflected tone as the cyclist rides. Assume the speed of sound in air is 337 m/s. Cyclist with bell Wall Figure 1: A cyclist with a bell riding towards a brick wall. The cyclist rings the bike's bell and the note is reflected off of the wall.

Answer 1

Part A.

given that beat frequency detected by sound meter is 13 Hz, So

f1 - f0 = 13 Hz

See that reflected frequency will have higher value since cyclisit is riding towards the stationary wall.

f1 = f0 + 13

f0 = frequecny of bell = 658 Hz

f1 = f_reflected = 658 + 13 = 671 Hz

Part B.

Using doppler's effect:

frequency received at the wall will be, when Source is moving towards the stationary observer:

f0' = f0*(V - Vo)/(V - Vs)

f0 = 658 Hz

V = Speed of Sound = 337 m/sec

Vs = speed of cyclist (Source) = ? m/sec

Vo = Speed of observer (wall) = 0 m/s

f0' = 658*(337 - 0)/(337 - Vs)

Now frequency reflected by wall will be

f1 = f0'*(V + Vo')/(V + Vs')

Vs' = Speed of source (wall) = 0 m/s (See that now wall has become source as sound is reflected from wall)

Vo' = speed of observer (cyclist) = Vs

f1 = f0'*(337 + Vs)/(337 + 0)

So, Using above value of f0'

f1 = 658*(337 + Vs)*(337 - 0)/[(337 - Vs)*(337 + 0)]

f1 = 658*(337 + Vs)/(337 - Vs)

Now given that beat frequency detected by sound meter is 13 Hz, So

f1 = 671 Hz

671 = 658*(337 + Vs)/(337 - Vs)

337*671 - 671*Vs = 658*337 + 658*Vs

Vs = 337*(671 - 658)/(671 + 658) = 3.2964 m/s

Vs = speed of bicyclist = 3.3 m/s

Part C.

Now frequency of sound heard by stationary observer will be:

frequency received at the wall will be:

f0' = 658*(337 - 0)/(337 - Vs)

f0' = 658*(337 - 0)/(337 - 3.2964) = 664.4998

f0' = 664.5 Hz

Now frequency heard by stationary observer will be:

f1' = f0'*(343 + Vo')/(343 + Vs')

Now see that in this case wall and observer are both stationary, So

Vo' = Vs' = 0

f1' = f0' = 664.5 Hz = frequency of reflected sound heard by stationary observer

f1 = 671 Hz = frequency of reflected sound heard by cyclist

So, beat frequency between them will be:

f_beat = f1 - f1'

f_beat = 671 - 664.5

f_beat = 6.5 Hz

Let me know if you've any query.