Find the point of destructive interference of two waves.

[HunterDX77M]

Homework Statement

Two coherent in-phase point sources of sound are located at the points (-2 m, 0 m) and (2 m, 0 m). If the wavelength of the sound is 0.9 m, at which of the following x values on the x-axis does destructive interference occur?

Homework Equations

Wave Equation:
y(x, t) = Acos(kx + ωt)

The Attempt at a Solution

I have no idea how to even start this one. Could someone just nudge me in the right direction? This is the last practice problem I have to do, but I am really stuck.

 

[cepheid]

Well, maybe start with this: what is the condition for destructive interference to occur?

 

[HunterDX77M]

Well, let's see. They have to be out of phase by 180 degrees. But the part that confuses me is that the problem says they have the same phase.

 

[SammyS]

Well, let's see. They have to be out of phase by 180 degrees. But the part that confuses me is that the problem says they have the same phase.

The sources are in phase.

However, the waves move and that's at a finite speed.

 

[Nickie]

I understand your frustration with this problem. Finding the point of destructive interference of two waves requires a good understanding of wave properties and equations. Let's break down the problem step by step to help you find the solution.

First, let's define some key terms. Coherent means that the two waves have the same frequency and are in phase, meaning they have the same starting point and are moving in sync. In-phase means that the two waves have the same amplitude and wavelength.

Next, let's recall the wave equation: y(x, t) = Acos(kx + ωt). This equation represents a wave with amplitude A, wavelength λ, and angular frequency ω. The variable k represents the wave number and is equal to 2π/λ.

Now, let's apply this equation to the problem at hand. The two point sources of sound are located at (-2 m, 0 m) and (2 m, 0 m), which means they are both emitting waves along the x-axis. The wavelength of the sound is given as 0.9 m, so we can calculate the wave number as k = 2π/0.9 = 7π/5.

To find the point of destructive interference, we need to determine where the two waves cancel each other out. This occurs when the amplitude of one wave is equal to the negative amplitude of the other wave. In other words, when the two waves are 180 degrees out of phase.

Using our wave equation, we can set the two waves equal to each other and solve for x:

Acos(kx + ωt) = -Acos(kx + ωt)

Since the frequencies and amplitudes are the same for both waves, we can cancel them out:

cos(kx + ωt) = -cos(kx + ωt)

Next, we can use the trigonometric identity cos(x) = -cos(x + π) to simplify the equation:

cos(kx + ωt) = cos(kx + ωt + π)

This means that the two waves will cancel each other out at any point where kx + ωt + π = 0. Solving for x, we get:

kx + ωt + π = 0

7πx/5 + ωt + π = 0

7πx/5 = -π

x = -