Solve Sound Wave Problem: Frequency of Sound in Air (336 m/s)

[Saitama]

Homework Statement

A source of sound S and a detector D are placed at some distance from one another. A big cradboard is placed near the detector and perpendicular to line SD as shown in figure. It is gradually moved away and it is found that the intensity changes from a maximum to a minimum as the board is moved through a distance of 20 cm. Find the frequency of the sound emitted. Velocity of sound in air is 336 m/s.

Homework Equations

The Attempt at a Solution

I have posted a similar problem in the past.
https://www.physicsforums.com/showthread.php?t=684072&highlight=sound+detector+wall

(The figure is present in the linked thread. I can't seem to attach it to the current thread )

I tried the current problem the following way:

Let the cardboard be initially at a distance $x$ from S and let D be at a distance $y$ from S. The path difference between the two waves reaching D is $2(x-y)$. For a maximum, the path difference must be $n\lambda$ i.e $2(x-y)=n\lambda\,\, (*)$ where n is a non-negative integer. When the cardboard is moved by 20 cm, the path difference is $2(x-y)-40$. For a minimum, this must be a equal to $(m+1/2)\lambda$ where m is a non-negative integer. Using (*), I get:
$$n\lambda-40=\left(m+\frac{1}{2}\right)\lambda$$
Solving for $\lambda$
$$\lambda=\frac{40}{n-m-1/2}$$
If I assume $n=1$ and $m=0$, I get the given answer i.e 420 Hz but if I assume some other values, I get different answers.

Any help is appreciated. Thanks!

 

[Tanya Sharma]

The key thing is that the path difference between a consecutive maxima and minima is λ/2 .

Now when the detector is moved 20 cm away ,the path difference introduced is 40 cm .

i.e 40 = λ/2 or λ = 80cm .

f=c/λ = 420 Hz .

 

[ehild]

See how you can attach an old picture

Or you can attach it again, by changing the name.

Tania is right, the keyword is "consecutive" n and m differ by one.

ehild

 

[Saitama]

Thanks a lot Tanya and ehild!

I had a similar dilemma with the other problems I am currently going through but now they are all solved, thanks! :)