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Chapter 15: Problem 33

(II) Show that the wave forms of Eqs. 13 and 15 satisfy the wave equation, Eq. \(16 .\) $$D(x, t)=A \sin \left[\frac{2 \pi}{\lambda}(x+v t)\right] (13 a)$$ $$=A \sin \left(\frac{2 \pi x}{\lambda}+\frac{2 \pi t}{T}\right) (13 b)$$ $$=A \sin (k x+\omega t) (13 b) $$ $$D(x, t)=D(x+v t) (15)$$ $$\frac{\partial^{2} D}{\partial x^{2}}=\frac{1}{v^{2}} \frac{\partial^{2} D}{\partial t^{2}} (16)$$

Short Answer

Expert verified
The wave forms satisfy the wave equation as \( k^2 = \frac{\omega^2}{v^2} \).

Step by step solution

01

Understanding the Variables

The given wave function is \( D(x, t) = A \sin(kx + \omega t) \). The variables involved are: \( k = \frac{2\pi}{\lambda} \), the wave number, and \( \omega = \frac{2\pi}{T} \), the angular frequency. The wave speed \( v \) is defined as \( v = \frac{\omega}{k} \).
02

Differentiate with Respect to x

Compute the first and second partial derivatives of \( D(x, t) = A \sin(kx + \omega t) \) with respect to \( x \): \[ \frac{\partial D}{\partial x} = A k \cos(kx + \omega t) \]\[ \frac{\partial^2 D}{\partial x^2} = -A k^2 \sin(kx + \omega t) \]
03

Differentiate with Respect to t

Compute the first and second partial derivatives of \( D(x, t) = A \sin(kx + \omega t) \) with respect to \( t \):\[ \frac{\partial D}{\partial t} = A \omega \cos(kx + \omega t) \]\[ \frac{\partial^2 D}{\partial t^2} = -A \omega^2 \sin(kx + \omega t) \]
04

Substitute into the Wave Equation

Substitute the second derivatives from Steps 2 and 3 into the wave equation \( \frac{\partial^2 D}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 D}{\partial t^2} \):\[ -A k^2 \sin(kx + \omega t) = \frac{1}{v^2} (-A \omega^2 \sin(kx + \omega t)) \]Since both sides include the factor \(-A \sin(kx + \omega t)\), divide both sides by this factor to obtain:\[ k^2 = \frac{\omega^2}{v^2} \]
05

Verify the Wave Speed Relationship

Using the wave speed definition \( v = \frac{\omega}{k} \), square both sides to check the consistency with the derived formula. This gives \( v^2 = \frac{\omega^2}{k^2} \), confirming that \( k^2 = \frac{\omega^2}{v^2} \) holds true. Thus, the wave form satisfies the wave equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
In mathematics and physics, partial derivatives are used to measure how a function changes when one of the input variables is varied and others are held constant. They are fundamental for understanding wave functions.
The wave function is often represented as a function of both position and time, such as \( D(x, t) = A \sin(kx + \omega t) \). Here, taking partial derivatives helps us analyze how this function changes as either \( x \) or \( t \) changes separately.
  • First partial derivative with respect to \( x \): This tells us how the wave function changes as we move through space. For our wave function, this derivative is \( \frac{\partial D}{\partial x} = A k \cos(kx + \omega t) \).
  • Second partial derivative with respect to \( x \): This gives us a measure of the wave's concavity in space and is \( \frac{\partial^2 D}{\partial x^2} = -A k^2 \sin(kx + \omega t) \).
  • First partial derivative with respect to \( t \): This shows the wave's change over time, given by \( \frac{\partial D}{\partial t} = A \omega \cos(kx + \omega t) \).
  • Second partial derivative with respect to \( t \): This examines the wave's acceleration over time and is \( \frac{\partial^2 D}{\partial t^2} = -A \omega^2 \sin(kx + \omega t) \).
Understanding these derivatives is crucial for solving the wave equation and evaluating a wave's behavior in different circumstances.
Wave Function
The wave function \( D(x, t) = A \sin(kx + \omega t) \) is central to wave mechanics. It describes the oscillation pattern of a wave moving through a medium over time.
  • Amplitude \( A \): This represents the maximum extent of a vibration or displacement of the wave. It's the peak value of the wave function.
  • Argument \( kx + \omega t \): The phase of the wave determines its position in the cycle of motion. Inside the sine function, this phase changes with both position \( x \) and time \( t \).
  • Sine function \( \sin \): This trigonometric function is used to model periodic oscillations, making it perfect for wave behavior. It indicates that the wave is continuous and repetitive.
In summary, the wave function is a mathematical expression defining how the wave behaves at every point in space and time. Understanding it allows for the visualization and prediction of wave patterns.
Angular Frequency
Angular frequency \( \omega \) is a cornerstone concept in oscillatory motion analysis. It quantifies how fast the wave oscillates at a given point in time.
  • Formula: It is defined as \( \omega = \frac{2\pi}{T} \), where \( T \) is the wave's period, or the time it takes for one complete cycle.
  • Role in the Wave Function: In the wave equation \( D(x, t) = A \sin(kx + \omega t) \), \( \omega t \) shows how much the wave has progressed at any time \( t \).
  • Relation to Frequency \( f \): Frequently, we compare angular frequency with regular frequency \( f \), where \( f = \frac{1}{T} \). Angular frequency is thus \( \omega = 2\pi f \).
Angular frequency ensures that we maintain consistency in wave analysis, particularly when converting between time-based wave behavior and radians per second dynamics.
Wave Number
Wave number \( k \) is a vital concept that measures wave properties in space. It represents the number of wave cycles within a unit distance.
  • Definition: Wave number is defined as \( k = \frac{2\pi}{\lambda} \), where \( \lambda \) is the wavelength, the distance over which the wave's shape repeats.
  • Role in Wave Function: In the wave function \( D(x, t) = A \sin(kx + \omega t) \), \( kx \) indicates how the wave evolves over space.
  • Spatial Frequency: Analogous to angular frequency, wave number quantifies the spatial frequency of the wave, providing a measure of how densely the waves are packed together along the axis.
Understanding wave number helps visualize how the wave propagates in space, crucial for applications across physics and engineering.

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Most popular questions from this chapter

(1) Suppose at \(t=0,\) a wave shape is represented by \(D=A \sin (2 \pi x / \lambda+\phi) ;\) that is, it differs from Eq. 9 by a constant phase factor \(\phi .\) What then will be the equation for a wave traveling to the left along the \(x\) axis as a function of \(x\) and \(t ?\)

(I) A violin string vibrates at \(441 \mathrm{~Hz}\) when unfingered. At what frequency will it vibrate if it is fingered one-third of the way down from the end? (That is, only two-thirds of the string vibrates as a standing wave.)

(II) In deriving Eq. \(2, \quad v=\sqrt{F_{\mathrm{T}} / \mu,}\) for the speed of a transverse wave on a string, it was assumed that the wave's amplitude \(A\) is much less than its wavelength \(\lambda\) . Assuming a sinusoidal wave shape \(D=A \sin (k x-\omega t),\) show via the partial derivative \(v^{\prime}=\partial D / \partial t \quad\) that the assumption \(A \ll \lambda\) implies that the maximum transverse speed \(v_{\text { max }}^{\prime}\) the string itself is much less than the wave velocity. If \(A=\lambda / 100\) determine the ratio \(v_{\max }^{\prime} / v.\) \(v=\sqrt{\frac{F_{\mathrm{T}}}{\mu}} \quad \quad \left[ \begin{array}{l}{\text { transverse }} \\ {\text { wave on a cord }}\end{array}\right]\) (2)

(I) Water waves approach an underwater "shelf" where the velocity changes from \(2.8 \mathrm{~m} / \mathrm{s}\) to \(2.5 \mathrm{~m} / \mathrm{s}\). If the incident wave crests make a \(35^{\circ}\) angle with the shelf, what will be the angle of refraction?

(II) A wire is composed of aluminum with length \(\ell_{1}=0.600 \mathrm{~m}\) and mass per unit length \(\mu_{\mathrm{}}=2.70 \mathrm{~g} / \mathrm{m}\) joined to a stecl section with length \(\ell_{2}=0.882 \mathrm{~m}\) and mass per unit length \(\mu_{2}=7.80 \mathrm{~g} / \mathrm{m} .\) This composite wire is fixed at both ends and held at a uniform tension of \(135 \mathrm{~N}\). Find the lowest frequency standing wave that can exist on this wire, assuming there is a node at the joint between aluminum and steel. How many nodes (including the two at the ends) does this standing wave possess?
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