Finding click frequency when echoes coincide with new clicks

[CrazyNeutrino]

Homework Statement

This is a problem from the Cambridge Natural Science Admissions Assesment.

A student carries out an experiment to measure the speed of sound. A loudspeaker that emits sound in all directions is placed between two buildings that are 128 m apart as shown. The student and loudspeaker are 48 m from one of the buildings.

The loudspeaker is connected to a signal generator that causes it to emit regular clicks. The student notices that each click results in two echoes, one from each building. The rate at which the clicks are produced is gradually increased from zero until each echo coincides with a new click being emitted by the loudspeaker.

What is the frequency of emission of clicks when this happens? (The speed of sound in air = 320 m/s)

Homework Equations

$$v=\frac{d}{t}$$
$$f=1/T$$

The Attempt at a Solution

The time taken between clicks must be equal to the time taken between the student hearing echoes at this frequency. $t_1= \frac{96}{320}$ and $t_2=\frac{160}{320}$

$$T_{click} = t_2 - t_1 = \frac{64}{320} = \frac{1}{5}$$
$$F=\frac{1}{T}=5 Hz$$

The correct answer is 10 Hz. Why?

 

[phyzguy]

In order to have both echoes arrive at the time of the next click, it is necessary that there are an integer number of clicks in both t1 and t2. That is there must be integers m and n such that m*t2 = n*t1. Since the frequency of clicks is slowly increased from zero, you are looking for the smallest integers for which this is satisfied. These are m=3 and n=5. Then Tclick = t1/3 = t2/5 = 32/320 = 1/10. You implicitly assumed m=1 and n=2, which doesn't work.

 

[Andrew Mason]

First of all, you are given the speed of sound so I am a bit puzzled by the your title.

To determine the frequency you need to work out the relationship between the number of click intervals that must occur between emission of the click and reception of the echo. If they are co-incident then there must be a whole number of intervals. You can easily determine that for the longer of the two paths (160 m). there is half a second interval between click emission and reception of an echo. For the shorter path, that interval is .3 seconds. So the question is how many clicks per second results in a whole number of intervals for each path. 2n clicks per second works for the long path but not for the short one (n is a whole number). What is the minimum even number of clicks per second that you need to have in order to give a whole number of intervals for the short path?

Another approach would be to take the ratio of intervals between the two paths for co-incident receptions and find the first whole number multiple of that ratio that results in a whole number to find the lowest value for n. Then use f = 2n

AM

 

[CrazyNeutrino]

Sorry about the title, the “measuring the speed of sound” part was added by a moderator for ‘not being descriptive enough’. Not my doing.