Problem 18 - Classical Dynamics of Particles and Systems

Question

Sketch the phase velocity \(V(k)\) and the group velocity \(U(k)\) for the propagation of waves along a loaded string in the range of wave numbers \(0 \leq k \leq \pi / d\). Show that \(U(\pi / d)=0,\) whereas \(V(\pi / d)\) does not vanish. What is the interpretation of this result in terms of the behavior of the waves?

Step-by-Step Solution

Answer

In summary, the phase velocity \(V(k)\) and group velocity \(U(k)\) for the propagation of waves along a loaded string within a range of wave numbers \(0 \leq k \leq \pi / d\) are given by: \[ V(k) = \frac{\sqrt{\frac{A}{\rho}} \left[\sin(kd) +\frac{Kr^{2}}{2A}\right]}{k} \] and \[ U(k) = d\sqrt{\frac{A}{\rho}}\cos(kd) \] We found that \(U(\pi / d) = 0\) and \(V(\pi / d) \neq 0\). This result indicates that wave packets (groups of waves) at maximum value of \(k\) do not propagate, as destructive interference prevents information transmission in the longitudinal direction. However, individual wave components continue to propagate due to a non-vanishing phase velocity.

Step 1: Compute phase velocity (V)

The phase velocity \(V\) is defined as the ratio of frequency \(\omega\) with respect to the wave number \(k\): \[ V(k) = \frac{\omega(k)}{k} \tag{2}\] Now, substitute equation \((1)\) into equation \((2)\) to obtain the V(k) relation: \[ V(k) = \frac{\sqrt{\frac{A}{\rho}} \left[\sin(kd) +\frac{Kr^{2}}{2A}\right]}{k} \tag{3}\]

Step 2: Compute group velocity (U)

The group velocity \(U\) is defined as the derivative of the frequency \(\omega\) with respect to the wave number \(k\): \[ U(k) = \frac{d\omega(k)}{dk} \tag{4}\] Now, differentiate equation \((1)\) with respect to \(k\) to obtain the U(k) relation: \[ U(k) = \frac{d}{dk}\left(\sqrt{\frac{A}{\rho}} \left[\sin(kd) +\frac{Kr^{2}}{2A}\right]\right) = \sqrt{\frac{A}{\rho}}\left[\cos(kd)\cdot d + 0\right] = d\sqrt{\frac{A}{\rho}}\cos(kd) \tag{5}\]

Step 3: Evaluate V(π/d) and U(π/d)

Plug \(k = \pi / d\) into equations \((3)\) and \((5)\) to evaluate V(π/d) and U(π/d): \[ V\left(\frac{\pi}{d}\right) = \frac{\sqrt{\frac{A}{\rho}} \left[\sin(\pi) +\frac{K(\pi^{2})}{2Ad^{2}}\right]}{\frac{\pi}{d}} \] Since \(\sin(\pi) = 0\), the equation simplifies to: \[ V\left(\frac{\pi}{d}\right) = \frac{\sqrt{\frac{A}{\rho}}\frac{K(\pi^{2})}{2Ad^{2}}}{\frac{\pi}{d}} \] Therefore, \(V(\pi / d)\) does not vanish: \[ V\left(\frac{\pi}{d}\right) \neq 0 \] Now, calculate the group velocity at \(k = \pi / d\): \[ U\left(\frac{\pi}{d}\right) = d\sqrt{\frac{A}{\rho}}\cos\left(\frac{\pi}{d}\cdot d\right) = d\sqrt{\frac{A}{\rho}}\cos(\pi) \] Since \(\cos(\pi) = -1\), the equation simplifies to: \[ U\left(\frac{\pi}{d}\right) = 0 \]

Step 4: Interpret the result

When the group velocity \(U(\pi / d) = 0\), it indicates that any wave packets (groups of waves) at the maximum value of \(k\) does not propagate, so information cannot propagate in the longitudinal direction effectively. This can be due to the destructive interference of individual components of the wave packets. On the other hand, the phase velocity \(V(\pi / d)\) does not vanish, meaning the individual wave components continue to propagate.

Key Concepts

Phase Velocity

Phase velocity is a fundamental aspect of wave propagation, referring to the speed at which a particular point of identical phase (say, a crest or a trough) moves through space. In the context of loaded string waves, the phase velocity, denoted as \( V(k) \), is determined by the relationship between the frequency \( \omega \) and the wave number \( k \), which is the spatial frequency of the wave. Mathematically, the phase velocity is expressed as \( V(k) = \frac{\omega(k)}{k} \).

For a loaded string, the mass distribution and the tension affect the speed of the wave, where \( \omega(k) \) is related to both wave number and the properties of the string such as tension \( A \), mass density \( \rho \), and other factors specific to how the string is loaded. Interestingly, at the maximum wave number ( \( k = \frac{\pi}{d} \) ), the phase velocity does not vanish, which suggests that individual wave components continue to move forward even if the collective motion or group velocity ceases.

Group Velocity

Group velocity describes the speed at which the overall shape of a wave's amplitude, known as the wave packet, travels. For physical waves on a loaded string, this relates to how quickly the energy or information is transmitted. The group velocity, indicated as \( U(k) \), is given by the derivative of the frequency with respect to the wave number: \( U(k) = \frac{d\omega(k)}{dk} \).

The significance of the group velocity becomes apparent when considering wave packets. If the group velocity is zero, which can occur at the wave number \( k = \frac{\pi}{d} \), it indicates that the wave packets do not move along the string. This situation could be indicative of an upper limit in the frequency that the string can support, or due to the unique characteristics of the wave's interference at that point.

Wave Number

The wave number, often denoted by \( k \), is a measure of the number of wave cycles per unit distance. It's a useful component in characterizing waves on a string, as it directly relates to their frequency and wavelength. We typically define the wave number mathematically as \( k = \frac{2\pi}{\lambda} \), where \( \lambda \) is the wavelength.

In the case of a loaded string, the wave number can vary from zero to a certain maximum value, beyond which the properties of the wave may change drastically. It is this range that influences both the phase and group velocities, as seen in the equations provided in the textbook solution. The higher the value of \( k \), the smaller the wavelength, which translates to how tightly packed the wave cycles are on the string.

Loaded String Waves

Loaded string waves refer to wave motions in a medium where the mass is not uniformly distributed. A string may be 'loaded' by adding extra mass at regular intervals, changing its normal characteristics. This loading influences the wave propagation, resulting in unique behaviors for both the phase and group velocities.

The loading of the string typically modifies its mass per unit length \( \rho \), and thus directly affects the frequency equation shown in the problem. Such modifications lead to interesting phenomena such as the non-zero phase velocity at the maximum wave number, while the group velocity can become zero, signifying no net transport of energy at this particular wave number. Understanding loaded string waves helps students grasp the complexity of wave behaviors in non-uniform media.