Question

a) Starting from the general wave equation (equation 15.9 ), prove through direct derivation that the Gaussian wave packet described by the equation \(y(x, t)=(5.00 m) e^{-0.1(x-5 t)^{2}}\) is indeed a traveling wave (that it satisfies the differential wave equation). b) If \(x\) is specified in meters and \(t\) in seconds, determine the speed of this wave. On a single graph, plot this wave as a function of \(x\) at \(t=0, t=1.00 \mathrm{~s}\), \(t=2.00 \mathrm{~s},\) and \(t=3.00 \mathrm{~s}\) c) More generally, prove that any function \(f(x, t)\) that depends on \(x\) and \(t\) through a combined variable \(x \pm v t\) is a solution of the wave equation, irrespective of the specific form of the function \(f\).

Short answer

Question: Prove that the given Gaussian wave packet represents a traveling wave and satisfies the wave equation. Determine the speed of the wave and plot it as a function of x at various times. Prove that any function that depends on x and t through a combined variable x±vt will satisfy the wave equation. Answer: 1. The Gaussian wave packet satisfies the wave equation, which confirms that it represents a traveling wave. The through deriving the second-order partial derivatives with respect to x and t, we can show that the wave packet satisfies the wave equation. 2. The speed of the wave is 5 m/s. To plot the wave as a function of x at various times, substitute the respective time value into the Gaussian wave packet equation and create a graph with x on the horizontal axis and y(x, t) on the vertical axis. Plot the wave functions for t=0, 1, 2, and 3 seconds. 3. Any function f(x, t) that depends on x and t through a combined variable x±vt will satisfy the wave equation. This is proven by deriving the second-order partial derivatives with respect to x and t, and then confirming that the function satisfies the wave equation.

Step-by-step solution

Part a: Prove that the Gaussian wave packet satisfies the wave equation

To show that the Gaussian wave packet is a traveling wave, we need to show that it satisfies the wave equation, defined by \(\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}\). The given Gaussian wave packet is \(y(x, t)=(5.00 m) e^{-0.1(x-5 t)^{2}}\). Let's derive the first and second-order partial derivatives with respect to x and t. The first-order partial derivatives with respect to x and t are: \(\frac{\partial y}{\partial x} = (-0.1)(5.00) (2)(x-5t)e^{-0.1(x-5 t)^{2}}\) \(\frac{\partial y}{\partial t} = (0.1)(5.00) (10) t e^{-0.1(x-5 t)^{2}}\) Now, find the second-order partial derivatives: \(\frac{\partial^2 y}{\partial x^2} = -0.1(5.00)(2)((2)(-0.1)(x-5t)^{2} - 1)e^{-0.1(x-5 t)^{2}}\) \(\frac{\partial^2 y}{\partial t^2} = -0.1(5.00)(10)((2)(-0.1)(x-5t)^{2} + 1)e^{-0.1(x-5 t)^{2}}\) We can now show that the given wave packet satisfies the wave equation: \(\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}\) Plug the second-order partial derivatives into the equation and simplify: \(-0.1(5.00)(2)((2)(-0.1)(x-5t)^{2} - 1)e^{-0.1(x-5 t)^{2}} = \frac{1}{v^2} (-0.1(5.00)(10)((2)(-0.1)(x-5t)^{2} + 1)e^{-0.1(x-5 t)^{2}})\) Cancel out the common terms: \((2)((2)(-0.1)(x-5t)^{2} - 1)e^{-0.1(x-5 t)^{2}} = \frac{1}{v^2} (10)((2)(-0.1)(x-5t)^{2} + 1)e^{-0.1(x-5 t)^{2}}\) Divide both sides by 2: \(((-0.1)(x-5t)^{2} - 1)e^{-0.1(x-5 t)^2} = \frac{1}{v^2} (5)((2)(-0.1)(x-5t)^{2} + 1)e^{-0.1(x-5 t)^{2}}\) This proves that the Gaussian wave packet satisfies the wave equation and is a traveling wave.

Part b: Determine the speed of the wave and plot it as a function of x at t = 0, 1, 2, and 3 seconds

From the Gaussian wave packet, we can observe that \(v = 5\,m/s\). To plot the wave as a function of x at various times, we substitute the respective time value into the Gaussian wave packet equation: At t=0 seconds: \(y(x,0)=(5.00\,m) e^{-0.1(x-5\cdot0)^{2}}=(5.00\,m) e^{-0.1 x^{2}}\) At t=1 second: \(y(x,1)=(5.00\,m) e^{-0.1(x-5\cdot1)^{2}}=(5.00\,m) e^{-0.1(x-5)^{2}}\) At t=2 seconds: \(y(x,2)=(5.00\,m) e^{-0.1(x-5\cdot2)^{2}}=(5.00\,m) e^{-0.1(x-10)^{2}}\) At t=3 seconds: \(y(x,3)=(5.00\,m) e^{-0.1(x-5\cdot3)^{2}}=(5.00\,m) e^{-0.1(x-15)^{2}}\) Now, you can create a graph of the Gaussian wave packet with x in meters on the horizontal axis and y(x, t) on the vertical axis. Plot the wave functions for t=0, 1, 2, and 3 seconds.

Part c: Prove that any function f(x, t) that depends on x and t through a combined variable x±vt is a solution of the wave equation

Let \(f(x,t) = F(x \pm vt)\). To show that this function is a solution of the wave equation, we will derive the second-order partial derivatives with respect to x and t. The first-order partial derivatives with respect to x and t are: \(\frac{\partial f}{\partial x} = F'(x \pm vt)\) \(\frac{\partial f}{\partial t} = \pm vF'(x \pm vt)\) Now, find the second-order partial derivatives: \(\frac{\partial^2 f}{\partial x^2} = F''(x \pm vt)\) \(\frac{\partial^2 f}{\partial t^2} = v^2F''(x \pm vt)\) Now, we can show that the function f(x,t) satisfies the wave equation: \(\frac{\partial^2 f}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}\) Plug the second-order partial derivatives into the equation: \(F''(x \pm vt) = \frac{1}{v^2} v^2F''(x \pm vt)\) Because \(F''(x \pm vt)\) is common to both sides of the equation, this proves that any function f(x,t) that depends on x and t through a combined variable x±vt will satisfy the wave equation, irrespective of the specific form of the function f.