Question

What is the displacement of the string as a function of x at time T/4, where T is the period of oscillation of the string? Express the displacement in terms of A,x , and k only. That is, evaluate w * T/4 and substitute it in the equation for y(x,t).

Answer 1

At time T/4, the displacement of a string undergoing sinusoidal wave motion is y(x, T/4) = -A cos(kx), dependent on the amplitude A, position x, and wave number k.

Step-by-step explanation:

The student is asking about a wave on a string and is looking to find the displacement of the string as a function of position x at a specific time T/4, where T is the period of the wave. Given that the displacement of a wave on a string can be modeled as y(x, t) = A sin (kx - ωt), where A is the amplitude, k is the wave number, and ω is the angular frequency. To find the displacement at time T/4, we'll substitute that time into the equation and make use of the fact that ωT = 2π, so at T/4, ωT/4 = π/2.

So, the displacement at time T/4 is y(x, T/4) = A sin(kx - π/2), which simplifies to y(x, T/4) = -A cos(kx). This shows that the displacement is solely dependent on position x and the amplitude A at this particular time slice.

Answer 2

The displacement of the string at time T/4 is given by the function y(x, T/4) = A cos(kx), where A is the amplitude, x is the position along the string, and k is the wave number.

Step-by-step explanation:

The displacement y(x,t) of the string as a function of x at time T/4, where T is the period of oscillation of the string, and the string's displacement is described by the function y(x, t) = A sin(kx - wt). To find the displacement at t = T/4, we substitute T/4 into the function for t. The angular frequency ω is given by 2π / T. Therefore, at t = T/4, the term ωt becomes 2πT / 4T or π/2. Substituting this value into our displacement equation, we have y(x, T/4) = A sin(kx - π/2), which simplifies to y(x, T/4) = A cos(kx), since sin(θ - π/2) = cos(θ).