Vibrations and the wave equation

[Tony11235]

An infinite string vibrates according to the homogenenous wave equation $$u_{tt}-u_{xx} = 0$$ with initial data given by $$u(x, 0) =f(x)$$ and $$u_{t}(x, 0) = g(x)$$ for -infinity<x<infinity where both f and g are smooth functions that are positive on the intervals -4<x<-3 and 2<x<3 and both zero everywhere else along the x-axis. A person stands at location x=0.

The question is during what intervals of time will the person notice the string vibrating? I know the interval is t=x to t=infinity but what is the process of getting x? Sorry if this is a stupid question.

 

[Nate810]

The process of getting x involves solving the wave equation. This can be done by using Fourier series or separation of variables. You would need to find the solution to the wave equation with initial data given by f and g. Once you have the solution to the wave equation, you can then determine when the person at x=0 will notice the string vibrating. This can be done by evaluating the solution at x=0 and finding the intervals of time for which the solution is not zero.

 

[quicknote]

I can provide a response to the given content. The wave equation is a fundamental equation that describes the behavior of waves in different systems, such as strings, fluids, and electromagnetic fields. In this specific case, we are dealing with an infinite string that is vibrating according to the homogeneous wave equation. This equation tells us that the second derivative of the displacement of the string with respect to time (u_tt) is equal to the second derivative of the displacement with respect to distance (u_xx).

The initial data provided in the problem gives us information about the initial displacement (f(x)) and initial velocity (g(x)) of the string at different points along the x-axis. From this, we can see that the string is initially at rest (g(x) = 0) and has a positive displacement (f(x) > 0) only on the intervals -4<x<-3 and 2<x<3. This means that the string is at rest at all other points along the x-axis.

The question asks us to determine the intervals of time during which a person standing at x=0 will notice the string vibrating. To answer this question, we need to understand the concept of wave propagation. When a disturbance (in this case, the initial displacement and velocity of the string) is applied to a medium (the string), it propagates through the medium in the form of a wave. The speed at which this wave propagates is determined by the properties of the medium, such as its density and tension.

In the case of an infinite string, the wave speed is given by the square root of the tension divided by the density of the string. This speed is constant and is denoted by the symbol c. So, the wave will propagate with a constant speed c along the string.

Now, to determine the time intervals during which the person will notice the string vibrating, we need to consider the position of the person (x=0) and the speed of the wave (c). The person will notice the string vibrating when the wave reaches their position. Since the person is standing at x=0, the wave will reach them at time t=x/c. This means that the person will notice the string vibrating for all time intervals from t=x/c to t=infinity.

In summary, the person will notice the string vibrating for all time intervals from t=x/c to t=infinity, where x is the distance of the person from the nearest point