Solve Nodal Wave Problem: Find Number of Nodal Line & Frequency Increase

[euphoriax]

Homework Statement

Two loudspeakers connected to the same source produce sound waves in plase with the same amplitude. The frequency of the sound is 215 Hz and the speed of the sound is 344 m/s. A girl finds that when she stands at a point that is 15.0m from one source and 19.0m from the other she hears no sound.
Find:
a) the number of the nodal line on which the girl is located.
b) the minimum increase in frequency which would enable the girl to hear a maximum sound at the same location

Homework Equations

l S₁-S₂ l = (n- 1/2) wavelength
v= f(wavelength)

The Attempt at a Solution

a)
f= 215 Hz
v= 344 m/s
S1= 15.0m
S2= 19.0m
wavelength?
n=?

v= f(wavelength)
wavelength= 344/ 215
wavelength= 1.6m

l S₁- S₂ l= (n-1/2)(wavelength)
n= (lS₁-S₂l)/(wavelength) +0.5
n= (l 15.0-19.0l)/1.6m + 0.5
n=3

b) I don't really understand this part of the question, or how to find it so any help would be very much appreciated.


Thanks!

 

[anathema]

Thank you for your interesting question. Let's break down the problem and solve it step by step.

a) To find the number of the nodal line on which the girl is located, we need to use the formula you have mentioned: l S1-S2 l = (n- 1/2) wavelength.

First, we need to calculate the wavelength using the formula v= f(wavelength). We know the speed of sound (v) is 344 m/s and the frequency (f) is 215 Hz. Therefore, the wavelength is 344/215 = 1.6 m.

Next, we can substitute the values in the formula l S1- S2 l = (n-1/2)(wavelength). We know S1=15.0m, S2=19.0m, and the calculated wavelength is 1.6m. So, we have:

l 15.0-19.0 l = (n-1/2)(1.6)
4.0 = (n-1/2)(1.6)
n-1/2 = 4.0/1.6
n-1/2 = 2.5
n = 2.5+0.5
n = 3

Therefore, the girl is located on the 3rd nodal line.

b) Now, to find the minimum increase in frequency that would enable the girl to hear a maximum sound at the same location, we need to understand the concept of nodal lines. Nodal lines are the points where two waves with the same frequency and amplitude cancel each other out, resulting in no sound. However, if we increase the frequency slightly, the two waves will no longer be in phase and will not cancel each other out at the same point. This means that the girl will be able to hear the maximum sound at the same location.

To calculate the minimum increase in frequency, we need to find the difference between the frequencies of two consecutive nodal lines. In this case, the girl is located on the 3rd nodal line, and we need to find the frequency difference between the 3rd and 4th nodal lines.

We know that the wavelength is 1.6m, and the distance between two consecutive nodal lines is half of the wavelength. Therefore, the distance between the 3rd and 4th nodal lines