Constructive Interference Problem with 2 Speakers

[infamous24]

Homework Statement

You are standing a distance d (1 m) in front of one of two identical speakers, being driven by the same source, that are a distance h (3 m) apart. You walk (starting at y=0) along a line parallel to the line of the two speakers. The speed of sound is 340 m/s and the frequency is 170 Hz.

Question: As you walk, how many times and where will you hear a maximum sound?

Homework Equations

ΔФ= 2λ/m + ΔФo

The Attempt at a Solution

I solved the above equation for the variables y and m, but I am unable to solve the equation to find what values of y where a maximum sound will be heard. My teacher suggested to make a plot of the path difference as a function of y using the given d and h and use the graph to determine the values of y which satisfy the maximum sound condition.

 

[Ouabache]

Welcome to PF forums! If you've already taken a look around, you will notice there are many interesting and informative topics around this site. And also people willing to offer their expertise and guidance on questions.

Your teacher's suggestion is a sound approach (no pun intended). Perhaps you can draw a picture of what the problem states, indicating the direction you are moving, relative to the speakers and be sure to show the relative dimensions along the axis between the speakers and then post it here. Also show the shape of a waveform, as it is radiated from each speaker.

 

[Moetasim]

However, I am not sure what values of y to plot and how to interpret the graph to find the solutions.

As a scientist, it is important to understand and interpret data in order to solve problems. In this case, the maximum sound condition can be achieved when the path difference between the two speakers is an integer multiple of the wavelength. This is known as constructive interference, where the sound waves from the two speakers reinforce each other.

To solve this problem, we can use the formula ΔФ = 2λ/m + ΔФo, where ΔФ is the path difference, λ is the wavelength, m is the integer multiple, and ΔФo is the initial phase difference. In this case, the initial phase difference is 0, since both speakers are driven by the same source.

First, we need to find the wavelength of the sound wave, which can be calculated using the formula λ = v/f, where v is the speed of sound and f is the frequency. Plugging in the given values, we get λ = 340/170 = 2 m.

Next, we can plot the path difference as a function of y, where y represents the distance you walk from the initial position. The path difference formula can be rewritten as ΔФ = 2y/h + ΔФo. We can choose a few values of y and plot them on the graph, such as y = 0, 1, 2, 3, etc. The graph should look like a straight line with a slope of 2/h.

Now, we need to determine at which values of y the path difference is an integer multiple of the wavelength. These values will correspond to the locations where constructive interference occurs. In this case, the path difference needs to be equal to an integer multiple of 2 m.

For example, if we choose y = 2, the path difference is 2(2)/3 + 0 = 4/3 m, which is not an integer multiple of 2 m. However, if we choose y = 3, the path difference becomes 2(3)/3 + 0 = 2 m, which is exactly one wavelength. This means that at y = 3, you will experience constructive interference and hear a maximum sound.

Similarly, we can choose other values of y and determine at which points constructive interference occurs. The number of times you will hear a maximum sound will depend on how far you walk and the