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Magazine Chapter 29: Problem 48
The displacement \(y\) at any point in a taut, flexible string depends on the distance \(x\) from one end of the string and the time \(t\) Show that \(y(x, t)=2 \sin 2 x \cos 4 t\) satisfies the wave equation \(\frac{\partial^{2} y}{\partial t^{2}}=a^{2} \frac{\partial^{2} y}{\partial x^{2}}\) with \(a=2.\)
Short Answer
Expert verified
The function \( y(x, t) = 2 \sin(2x) \cos(4t) \) satisfies the wave equation with \( a = 2 \).
Step by step solution
01
Identify the Given Equation and Parameters
We are given the function \( y(x, t) = 2 \sin(2x) \cos(4t) \) and need to show it satisfies the wave equation \( \frac{\partial^{2} y}{\partial t^{2}}=a^{2} \frac{\partial^{2} y}{\partial x^{2}} \) with \( a = 2 \). This sets up our targets for the derivatives required.
02
Compute the First Partial Derivative with Respect to Time
Calculate the first partial derivative of \( y(x, t) \) with respect to \( t \):\[ \frac{\partial y}{\partial t} = \frac{\partial}{\partial t}(2 \sin(2x) \cos(4t)) = 2 \sin(2x) \cdot (-4 \sin(4t)) = -8 \sin(2x) \sin(4t) \]
03
Compute the Second Partial Derivative with Respect to Time
Calculate the second partial derivative of \( y(x, t) \) with respect to \( t \):\[ \frac{\partial^2 y}{\partial t^2} = \frac{\partial}{\partial t}(-8 \sin(2x) \sin(4t)) = -8 \sin(2x) \cdot 4 \cos(4t) = -32 \sin(2x) \cos(4t) \]
04
Compute the First Partial Derivative with Respect to Space
Calculate the first partial derivative of \( y(x, t) \) with respect to \( x \):\[ \frac{\partial y}{\partial x} = \frac{\partial}{\partial x}(2 \sin(2x) \cos(4t)) = 4 \cos(2x) \cos(4t) \]
05
Compute the Second Partial Derivative with Respect to Space
Calculate the second partial derivative of \( y(x, t) \) with respect to \( x \):\[ \frac{\partial^2 y}{\partial x^2} = \frac{\partial}{\partial x}(4 \cos(2x) \cos(4t)) = -8 \sin(2x) \cos(4t) \]
06
Verify the Wave Equation
Substitute the second derivatives into the wave equation and verify:The left side is:\[ \frac{\partial^2 y}{\partial t^2} = -32 \sin(2x) \cos(4t) \]The right side is:\[ a^2 \frac{\partial^2 y}{\partial x^2} = 4 \cdot (-8 \sin(2x) \cos(4t)) = -32 \sin(2x) \cos(4t) \]Since the left side equals the right side, the function satisfies the wave equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Partial Derivatives
Partial derivatives are a fundamental concept in calculus, particularly when dealing with functions of multiple variables like the wave function given in our exercise. The idea is to see how a function changes as one of its variables changes, keeping all other variables constant.
In the context of the wave equation, partial derivatives allow us to examine the change in displacement of the string over time (with respect to time) and across its length (with respect to space).
For instance:
In the context of the wave equation, partial derivatives allow us to examine the change in displacement of the string over time (with respect to time) and across its length (with respect to space).
For instance:
- To find how the displacement changes over time, we compute the partial derivative of the wave function with respect to time, denoted as \(\frac{\partial y}{\partial t}\).
- To see how the displacement subsequently varies along the string's length, we compute \(\frac{\partial y}{\partial x}\), the partial derivative with respect to spatial variable \(x\).
Exploring the Wave Function
A wave function is a mathematical description of the spatial and temporal dynamics of a wave. In physics, it helps us model natural phenomena such as light or sound waves, and in this exercise, the vibrations on a string.
The given wave function, \(y(x, t) = 2 \sin(2x) \cos(4t)\), describes how the position of every point on the string changes over time and space. Here:
The given wave function, \(y(x, t) = 2 \sin(2x) \cos(4t)\), describes how the position of every point on the string changes over time and space. Here:
- \(2\) is a constant factor affecting the amplitude or the maximum displacement of the wave.
- The sine function, \(\sin(2x)\), modulates the spatial characteristics, meaning it governs the displacement at different points along the string.
- The cosine function, \(\cos(4t)\), influences how this displacement varies with time.
Displacement in a String
Displacement in a string refers to how far some part of the string has deviated from its equilibrium (or rest) position at a given time. Understanding displacement is vital for analyzing wave phenomena.In the context provided, the displacement is given by the function \(y(x, t)\). It shows two crucial influences:
- How far each point on the string moves perpendicular to the string's normal position (affected by \(\sin(2x)\)).
- How this perpendicular movement changes as time progresses (shaped by \(\cos(4t)\)).
Applying Calculus in the Wave Equation
Calculus, especially differential calculus, is indispensable for tackling problems involving the wave equation. Through calculus, we compute derivatives that help us understand the change in the wave function over time and space.
In the case of the wave equation \(\frac{\partial^2 y}{\partial t^2} = a^2 \frac{\partial^2 y}{\partial x^2}\):
In the case of the wave equation \(\frac{\partial^2 y}{\partial t^2} = a^2 \frac{\partial^2 y}{\partial x^2}\):
- The term \(\frac{\partial^2 y}{\partial t^2}\) represents the second derivative with respect to time, outlining how the rate of change of the string's displacement accelerates over time.
- The term \(a^2 \frac{\partial^2 y}{\partial x^2}\) is similar but focuses on spatial changes, emphasizing the curvature of the string.
