Show that if f(x) is a function with a continuous second...

Show that if $f(x)$ is a function with a continuous second derivative, then $f(x-c t)$ and $f(x+c t)$ are both solutions of the wave equation of exercise $43,$ If $x$ represents position and $t$ represents time, explain why $c$ can be interpreted as the velocity of the wave.

Key Concepts

Wave Equation

The wave equation is a second-order partial differential equation generally written as u_tt = c^2 u_xx, where u(x, t) represents the wave function, t is time, x is position, and c denotes the speed of wave propagation. It serves as a mathematical model for various types of waves, such as sound and water waves.

Traveling Wave Solutions

Functions of the form f(x - ct) or f(x + ct) are examples of traveling wave solutions. They represent waves that maintain a fixed shape while moving at a constant velocity. The form of these functions indicates that the waveform is simply shifted along the x-axis over time, effectively capturing motion without deformation.

Chain Rule in Differentiation

To verify that f(x - ct) and f(x + ct) are solutions to the wave equation, the chain rule is applied to differentiate composite functions. This process involves computing the first and second derivatives with respect to both x and t, which reveals the relationship between the time and spatial derivatives and confirms that the wave equation is satisfied.

Interpretation of c as Wave Velocity

The constant c appears in the arguments of the functions as a factor of time multiplied by c, which implies that any fixed point on the function f moves a distance c units in one unit of time. This direct proportionality between time and displacement clearly establishes c as the velocity at which the wave propagates through space.

Transcript

00:02 Okay, this question gives us this arbitrary solution to the wave equation, where it says we just have some function f of x that has continuous derivatives, but now we plug in f of x plus or minus ct, and it says that this is a solution to the wave equation.

00:26 So we're just going to show this.

00:28 So if we take an x derivative, that's equal to f prime of x plus or minus c t times the chain roll factor of one if we're differentiating with respect to x or if we do it again we get f double prime of x plus or minus c t times one again so here's our expression for our second x derivative and again we don't have to worry about using partial derivative notation here on the right side because it says we're starting from a function f of x and then plugging in x plus or minus ct here.

01:19 So we really just have a chain rule going on and we're just changing what variable we're differentiating with respect to.

01:27 So now we're going to do the same thing but for the time derivative.

01:32 So if we take one time derivative, we still get f prime of x plus or minus ct, but now we have to worry about the chain rule factor, which is going to be either a plus or a minus c.

01:47 And then doing it again, we get f double prime of x plus or minus c t times now we have plus or minus c, but we get another one.

02:01 So we're squaring it.

02:02 Or gtt is equal to c squared, because the plus or minus is going to just become positive always, if we square it, times f double prime of x plus or minus ct.

02:24 And now if we look here, we see that gtt is just c squared off of our x partial.

02:36 So now if we look at this, we see that gtt is equal to c squared times gxx.

02:58 And that's the wave equation...

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