The Wave Equation Part 1: Vibrations in 1D

In the previous tutorial, we learned about Fourier transforms and how they can be used to identify what frequencies are present in a function and how they are especially useful for picking out the frequency components of sums of sine waves.

We will now study the partial differential equation that models wave propagation, and that is the aptly named wave equation.

The wave equation is a PDE that has the general form Laplacian of u equals one over c squared times the second partial derivative of u with respect to time.

Recall that the Laplacian operator can be expressed as a sum of second partial derivatives with respect to position, which in 3D Cartesian coordinates are x, y, and z.

So in the most general case, the wave equation is a PDE with four independent variables: x, y, z, and time t.

We will initially focus on the 1D case of the wave equation, and that will be uxx equals one over c squared times utt, where c is a constant representing the speed of the waves, and we can verify that the wave equation is a hyperbolic PDE, unlike the elliptic Laplace equation we looked at previously.

We will show two ways to solve the wave equation, first using separation of variables and second using Fourier transforms, both of which we covered in the previous tutorials.

For separation of variables, we assume that our solution u of x, t is the product capital X of x times capital T of t.

Substituting in and separating the variables leads to two Sturm-Liouville problems, and if we choose negative lambda squared as our separation constant, we will get trigonometric solutions for each of x and t, which is usually going to be the better choice for describing waves.

Without any boundary conditions, we can accept any value of lambda whatsoever, and the solution will be valid.

Note that in the context of waves, it might be better to use the wave number k instead of lambda, since lambda is typically used to mean wavelength.

For our function T of t, we can replace k times c with omega, which is the angular frequency of the wave.

Note that k and omega are not independent.

They are in a ratio defined by the wave speed c, which is a constant that was part of the original PDE.

When multiplied together, the functions x and t correspond to traveling waves, which we can more easily see by using the product to sum trig identities.

We can now interpret our solution as a wave in the x direction that has a time-dependent phase shift, always propagating through space at that fixed speed of c, which we will now call the phase velocity of the wave.

If our boundary conditions are such that the wave is constrained to have nodes at fixed points, such as the ends of a vibrating string, then we could identify the allowable frequencies, which are known as normal modes in the context of vibrations, and sum over those as shown here.

We can also interpret our answer as a Fourier series where the coefficients A and C are the terms of the Fourier series of some function of x plus or minus ct.

The roots of these functions, which are the lines x plus or minus ct equals zero when plotted in xt space, are called the characteristic lines of the wave, and these lines can be interpreted as the directions along which the information in the system can propagate.

We should also note that if the conditions are such that we find the constants A and C to be equal, our solution becomes a standing wave, which does not propagate through space at all, instead oscillating with fixed nodes or places-

In the previous tutorial, we learned about Fourier transforms and how they can be used to identify what frequencies are present in a function and how they are especially useful for picking out the frequency components of sums of sine waves.

We will now study the partial differential equation that models wave propagation, and that is the aptly named wave equation.

The wave equation is a PDE that has the general form Laplacian of u equals one over c squared times the second partial derivative of u with respect to time.

Recall that the Laplacian operator can be expressed as a sum of second partial derivatives with respect to position, which in 3D Cartesian coordinates are x, y, and z.

So in the most general case, the wave equation is a PDE with four independent variables: x, y, z, and time t.

We will initially focus on the 1D case of the wave equation, and that will be uxx equals one over c squared times utt, where c is a constant representing the speed of the waves, and we can verify that the wave equation is a hyperbolic PDE, unlike the elliptic Laplace equation we looked at previously.

We will show two ways to solve the wave equation, first using separation of variables and second using Fourier transforms, both of which we covered in the previous tutorials.

For separation of variables, we assume that our solution u of x, t is the product capital X of x times capital T of t.

Substituting in and separating the variables leads to two Sturm-Liouville problems, and if we choose negative lambda squared as our separation constant, we will get trigonometric solutions for each of x and t, which is usually going to be the better choice for describing waves.

Without any boundary conditions, we can accept any value of lambda whatsoever, and the solution will be valid.

Note that in the context of waves, it might be better to use the wave number k instead of lambda, since lambda is typically used to mean wavelength.

For our function T of t, we can replace k times c with omega, which is the angular frequency of the wave.

Note that k and omega are not independent.

They are in a ratio defined by the wave speed c, which is a constant that was part of the original PDE.

When multiplied together, the functions x and t correspond to traveling waves, which we can more easily see by using the product to sum trig identities.

We can now interpret our solution as a wave in the x direction that has a time-dependent phase shift, always propagating through space at that fixed speed of c, which we will now call the phase velocity of the wave.

If our boundary conditions are such that the wave is constrained to have nodes at fixed points, such as the ends of a vibrating string, then we could identify the allowable frequencies, which are known as normal modes in the context of vibrations, and sum over those as shown here.

We can also interpret our answer as a Fourier series where the coefficients A and C are the terms of the Fourier series of some function of x plus or minus ct.

The roots of these functions, which are the lines x plus or minus ct equals zero when plotted in xt space, are called the characteristic lines of the wave, and these lines can be interpreted as the directions along which the information in the system can propagate.

We should also note that if the conditions are such that we find the constants A and C to be equal, our solution becomes a standing wave, which does not propagate through space at all, instead oscillating with fixed nodes or places-