Question

Show that the function satisfies the wave equation \(\partial^{2} z / \partial t^{2}=c^{2}\left(\partial^{2} z / \partial x^{2}\right)\). \(z=\cos (4 x+4 c t)\)

Short answer

Yes, the function \(z=\cos (4x + 4ct)\) satisfies the wave equation because its second derivatives with respect to time and space fulfill the condition \(\partial^{2} z / \partial t^{2}=c^{2}(\partial^{2} z / \partial x^{2})\).

Step-by-step solution

Calculate the second derivative of z with respect to t

The first thing to do is to compute the second derivative of \(z\) with respect to \(t\). The function z is a composite function of the form \(\cos(u)\), with \(u = 4x + 4ct\). The derivative of \(\cos(u)\) is \(-\sin(u)\) and the derivative of \(u\) with respect to \(t\) is \(4c\). This implies that the first derivative of \(z\) with respect to \(t\) is obtained by applying the chain rule as follows: \[\partial z/ \partial t = -\sin(4x + 4ct) \times (4c)\]which simplifies to \[\partial z/ \partial t = -4c \sin(4x + 4ct)\]The second derivative of \(z\) with respect to \(t\) is again the derivative of the outcome, applying the chain rule a second time yields:\[\partial^{2} z / \partial t^{2} = -16c^{2} \cos(4x + 4ct) = -16c^{2}z\]

Calculate the second derivative of z with respect to x

Now we find the second derivative of z with respect to x. As in Step 1, we again have \(u = 4x + 4ct\), and the derivative of \(u\) with respect to \(x\) is 4. So the first derivative of \(z\) with respect to \(x\) is: \[\partial z/ \partial x = -\sin(4x + 4ct) \times 4\]which simplifies to \[\partial z/ \partial x = -4\sin (4x + 4ct)\]Similarly, taking the derivative again gives us the second derivative of \(z\) with respect to \(x\), which is: \[\partial^{2} z / \partial x^{2} = -16\cos (4x + 4ct) = -16z\]

Verify that the original equation holds

Substitute the second derivatives of \(z\) into the equation \(\partial^{2} z / \partial t^{2}=c^{2}(\partial^{2} z / \partial x^{2})\). The equation thus becomes: \[-16c^{2}z = c^{2}(-16z)\]Solving the equation shows that both sides are equal, thus confirming that the function \(z = \cos(4x + 4ct)\) does indeed satisfy the wave equation.

Key concepts

Partial Derivatives

Partial derivatives are essential when dealing with functions of multiple variables, as they show how a function changes as each variable changes, holding the others constant. Take, for example, the function z = \( \textrm{cos}(4x + 4ct) \) . To find how z changes with respect to time t while keeping x constant, we take the partial derivative of z with respect to t. This process involves treating x as if it were a constant, and differentiating z solely with respect to t. Similarly, when we need the rate of change of z with respect to x, we treat t as a constant and differentiate with respect to x. This step-by-step approach facilitates understanding more complex multi-variable functions and how they interact within an equation like the wave equation.

Chain Rule

The chain rule is a fundamental tool in calculus for tackling the derivatives of composite functions. When a function z is composed of another function u, such as u = 4x + 4ct in our wave equation example, the chain rule allows us to differentiate z in steps. First, we differentiate the outer function (here, the cosine function) with respect to u, and then multiply the result by the derivative of the inner function u with respect to our variable of interest (either x or t). This method provides clarity in the calculation process, especially when dealing with complex functions, ensuring that each part of the function is accounted for in the differentiation process.

Second Derivative

In the realm of calculus, the second derivative plays a crucial role in understanding the curvature of a function, offering insights into its concavity and points of inflection. By taking the derivative of a derivative, the second derivative measures how the rate of change of a quantity is itself changing. In our exercise, the function z with respect to both t and x is subjected to this process. The second derivative tells us how the slope of the tangent line to the curve changes as we move along the curve. This is vital in physics and engineering applications such as the wave equation, where the second derivatives relate to acceleration and curvature of waves.

Cosine Function

The cosine function, commonly denoted as \( \cos \), is a trigonometric function that oscillates between -1 and 1. It is periodic, making it ideal for representing wave-like phenomena. In the case of our wave equation exercise, z is expressed as a cosine function of both x and t, \( z = \cos(4x + 4ct) \). This formulation captures the wave's spatial (with x) and temporal (with t) behavior. The differential properties of the cosine function, namely cycling through the sequence of derivatives from cosine to -sine to -cosine to sine and back to cosine, facilitate the analysis and verification that the wave equation holds for the function z.