Question

When two pure tones with similar frequencies combine to produce beats, the result is a train of wave packets. That is, the sinusoidal waves are partially localized into packets. Suppose two sinusoidal waves of equal amplitude \(A\), traveling in the same direction, have wave numbers \(\kappa\) and \(\kappa+\Delta \kappa\) and angular frequencies \(\omega\) and \(\omega+\Delta \omega,\) respectively. Let \(\Delta x\) be the length of a wave packet, that is, the distance between two nodes of the envelope of the combined sine functions. What is the value of the product \(\Delta x \Delta \kappa ?\)

Short answer

Answer: The product of the length of a wave packet and the difference in wave numbers is 4nπ, where n is an integer.

Step-by-step solution

Write the sinusoidal wave equations

The two sinusoidal waves can be represented by the following equations: \(y_1 = A\sin(\kappa x - \omega t)\) \(y_2 = A\sin((\kappa+\Delta\kappa) x - (\omega+\Delta\omega) t)\)

Combine the sinusoidal waves

Next, we can add the sinusoidal waves together to create a combined wave function: \(Y(X,t) = y_1 + y_2 = A(\sin(\kappa x - \omega t) + \sin((\kappa+\Delta\kappa) x - (\omega+\Delta\omega) t))\)

Calculate the envelope of the combined sine functions

In order to find the envelope, we will use the trigonometric identity for the sum of two sinusoids: \(\sin(\alpha) + \sin(\beta) = 2\sin\Big(\frac{\alpha + \beta}{2}\Big)\cos\Big(\frac{\alpha - \beta}{2}\Big)\) Applying this identity to our combined wave function, we have: \(Y(X,t) = 2A\sin\Big(\frac{(\kappa x - \omega t) + ((\kappa+\Delta\kappa) x - (\omega+\Delta\omega) t)}{2}\Big)\cos\Big(\frac{(\kappa x - \omega t) - ((\kappa+\Delta\kappa) x - (\omega+\Delta\omega) t)}{2}\Big)\)

Simplify and identify the envelope function

Let's simplify the combined wave function first: \(Y(X,t) = 2A\sin\Big(\frac{4\omega t - 4\kappa x + 2\Delta \omega t - 2\Delta \kappa x}{4}\Big)\cos\Big(\frac{-2\omega t + 2\kappa x + 2\Delta \omega t - 2\Delta \kappa x }{4}\Big)\) After canceling out the common factors and reordering, we get: \(Y(X,t) = 2A\sin\Big(\frac{-\Delta\kappa x + \Delta\omega t}{2}\Big)\cos\Big(\frac{\kappa x - \omega t}{2}\Big)\) The envelope of the combined sine functions is given by the absolute value of the cosine term: \(E(X,t) = |2A\cos\Big(\frac{\kappa x - \omega t}{2}\Big)|\)

Find the distance between the envelope nodes

The distance between the nodes of the envelope, \(\Delta x\), is the distance between two consecutive points where the envelope has zero amplitude. This means that the cosine term equals zero: \(\cos\Big(\frac{\kappa x - \omega t}{2}\Big)=0\) Solving for \(x\), we get: \(\frac{\kappa x - \omega t}{2}=2n\pi\) \(\Delta x = \frac{4n\pi}{\kappa}\) where \(n\) is an integer.

Find the product of \(\Delta x\) and \(\Delta \kappa\)

Finally, we can find the product of the distance between the envelope nodes and the difference in wave numbers: \(\Delta x \Delta \kappa = \frac{4n\pi}{\kappa} \Delta \kappa\) \(= 4n\pi\) The product of \(\Delta x\) and \(\Delta \kappa\) is simply \(4n\pi\), where \(n\) is an integer.

Key concepts

Understanding Sinusoidal Wave Equations

Sinusoidal waves, a fundamental concept in physics and engineering, are elegantly described by sinusoidal wave equations. These mathematical expressions capture the oscillating nature of waves as they propagate through space and time.

Consider a wave, characterized by its amplitude, wavelength, frequency, and phase. The general equation for a sinusoidal wave is given by: \[\begin{equation}y = A \times \text{sin}(kx - \text{\omega} t + \text{\phi}),\end{equation}\] where:\begin{itemize}\item \(A\) denotes the amplitude, the peak value of the wave.\item \(k\), the wave number, relates to the wavelength \(\lambda\) by \(k = \frac{2\pi}{\lambda}\).\item \(\omega\), the angular frequency, is associated with the conventional frequency \(f\) through \(\omega = 2\pi f\).\item \(\phi\) represents the phase shift, which determines the wave's starting position.\end{itemize}When you stumble upon two waves, as in our exercise example, adding their equations together can reveal much about their combined behavior. By understanding these fundamentals clearly, you can grasp the complexities of wave interaction and superposition, setting the stage for further exploration of wave phenomena like interference patterns or, as in this case, the formation of wave packets.

Superposition of Waves

The principle of superposition is a cornerstone of wave theory. It states that when two or more waves meet, the resultant displacement at any point and time is the algebraic sum of the displacements due to the individual waves. This principle can be used to explain a wide range of phenomena including interference patterns and the formation of standing waves.

Our exercise exemplified this with two sinusoidal waves being added together to form a more complex wave. By using the superposition principle:\[\begin{equation}Y(X, t) = y_1 + y_2,\end{equation}\] we combine the individual wave functions to obtain the resultant wave. This interaction can either be constructive, where the waves reinforce each other, or destructive, where they cancel each other out, creating an interesting pattern of peaks and nulls - the wave packet's envelope. Essentially, this combined wave function is a snapshot of wave interference and offers insight into the nature of wave energy distribution in space. Whether it's sound, light, or water waves, the superposition of waves serves as an integral concept for explaining complex wave behaviors.

Beats in Wave Motion

When dealing with beats, one delves into the fascinating interplay between waves of nearly identical frequencies. Beats emerge as a result of the superposition principle and are manifest as periodic fluctuations in intensity of the resultant wave.

An everyday example is when two musical notes of similar, but not identical, frequencies are played simultaneously. The interference between the two sound waves produces a beat frequency that is the difference between the two original frequencies. Mathematically, this can be expressed as:\[\begin{equation}f_{\text{beat}} = |f_{1} - f_{2}|,\end{equation}\] where \(f_{1}\) and \(f_{2}\) are the frequencies of the two combining waves. Our exercise, with its elegant sequence of steps, showcases this concept by demonstrating the creation of wave packets as a physical representation of beats in wave motion. Here, the distance between the envelopes' nodes, \(\Delta x\), intertwines with the change in wave number, \(\Delta \kappa\), to depict the spatial aspect of beats. Together, they fulfill the role of visually illustrating the occurrence of beats in the waveforms we explored.