A string with linear mass density μ = 0.0279 kg/m under a tension of T = 183 N is oriented in the x-direction. Two transverse waves of equal amplitude and with a phase angle of zero (at t = 0) but with different frequencies (ω = 3420 rad/s and ω/3 = 1140 rad/s) are created in the string by an oscillator located at x = 0. The resulting waves, which travel in the positive x-direction, are reflected at a distant point, so there is a similar pair of waves traveling in the negative x-direction. Find the values of x at which the first two nodes in the standing wave are produced by these four waves.

Question

A string with linear mass density μ = 0.0279 kg/m under a tension of T = 183 N is oriented in the x-direction. Two transverse waves of equal amplitude and with a phase angle of zero (at t = 0) but with different frequencies (ω = 3420 rad/s and ω/3 = 1140 rad/s) are created in the string by an oscillator located at x = 0. The resulting waves, which travel in the positive x-direction, are reflected at a distant point, so there is a similar pair of waves traveling in the negative x-direction. Find the values of x at which the first two nodes in the standing wave are produced by these four waves.

Answer 1

To find the values of x at which the first two nodes in the standing wave are produced by these four waves, we need to understand the concept of standing waves and nodes.

A standing wave is formed when two waves with the same frequency and amplitude traveling in opposite directions interfere with each other. Nodes are the points on the wave where the displacement is always zero.

In this scenario, we have two waves traveling in the positive x-direction and two waves traveling in the negative x-direction. The two waves with frequencies ω = 3420 rad/s and ω/3 = 1140 rad/s combine to form a standing wave.

To find the nodes, we can use the following formula:

λ = 2L/n

where λ is the wavelength, L is the length of the string, and n is the number of nodes.

From the given information, the tension T, and the linear mass density μ of the string, we can calculate the speed of the waves using the formula:

v = √(T/μ)

Once we have the speed of the waves, we can calculate the wavelength using the formula:

v = ωλ

Now, let's calculate the speed of the waves:

v = √(T/μ)
v = √(183 N / 0.0279 kg/m)
v ≈ 61.823 m/s

Next, let's calculate the wavelength for each wave:

λ1 = v / ω
λ1 = 61.823 m/s / 3420 rad/s
λ1 ≈ 0.018 m

λ2 = v / (ω/3)
λ2 = 61.823 m/s / (1140 rad/s / 3)
λ2 ≈ 0.162 m

Now, we can find the x-values for the first two nodes using the formula:

x = m * λ / 2

where m is the mode (1 for the first node, 2 for the second node), and λ is the wavelength.

For the first node:

x1 = 1 * λ1 / 2
x1 = 0.018 m / 2
x1 ≈ 0.009 m

For the second node:

x2 = 1 * λ2 / 2
x2 = 0.162 m / 2
x2 ≈ 0.081 m

So, the values of x at which the first two nodes in the standing wave are produced by these four waves are approximately 0.009 m and 0.081 m.