Show that the displacement D(x,t) = cx² + dt², where c and d are constants, is a solution to the wave equation. Then find an expression in terms of c and d for the wave speed.

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Step 1: Recall the wave equation, which is typically written as ∂²D/∂t² = v² ∂²D/∂x², where D(x,t) is the displacement, v is the wave speed, and ∂²D/∂t² and ∂²D/∂x² are the second partial derivatives of D with respect to time and position, respectively.

Step 2: Compute the second partial derivative of D(x,t) = cx² + dt² with respect to time (t). Since the term cx² does not depend on t, its derivative with respect to t is zero. For the term dt², the second derivative with respect to t is 2d.

Step 3: Compute the second partial derivative of D(x,t) = cx² + dt² with respect to position (x). Since the term dt² does not depend on x, its derivative with respect to x is zero. For the term cx², the second derivative with respect to x is 2c.

Step 4: Substitute the computed second derivatives into the wave equation ∂²D/∂t² = v² ∂²D/∂x². This gives 2d = v² * 2c. Simplify this equation to find the relationship between v, c, and d.

Step 5: Solve for the wave speed v in terms of c and d. From the equation 2d = v² * 2c, divide both sides by 2c to isolate v², yielding v² = d/c. Take the square root of both sides to find v = √(d/c).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Wave Equation

The wave equation is a second-order partial differential equation that describes the propagation of waves through a medium. It is typically expressed as ∂²D/∂t² = v²∂²D/∂x², where D is the displacement, t is time, x is position, and v is the wave speed. Understanding this equation is crucial for analyzing wave behavior and determining if a given function, like D(x,t), satisfies the conditions of wave propagation.

Displacement Function

In the context of wave mechanics, the displacement function D(x,t) represents the position of points in a medium as a function of both space (x) and time (t). The form D(x,t) = cx² + dt² indicates that the displacement is a combination of spatial and temporal components, where c and d are constants. Analyzing this function helps in verifying if it meets the criteria set by the wave equation.

Wave Speed

Wave speed is the rate at which a wave propagates through a medium and is denoted by v in the wave equation. It can be derived from the relationship between the second derivatives of the displacement function with respect to time and space. In this case, finding the wave speed involves substituting the displacement function into the wave equation and solving for v in terms of the constants c and d.