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 value of the product ΔxΔκ is 2nπ, where n is an integer representing the nth node.

Step-by-step solution

Define the waves

Let's represent the two sinusoidal waves as follows: $$y_1 = A \cos (\kappa x - \omega t)$$ $$y_2 = A \cos ((\kappa+\Delta \kappa) x - (\omega+\Delta \omega) t)$$ where \(y_1\) and \(y_2\) represent the displacements of the waves, \(A\) is the amplitude, \(\kappa\) and \(\kappa+\Delta\kappa\) are the respective wave numbers, and \(\omega\) and \(\omega+\Delta\omega\) are the respective angular frequencies.

Add the two waves

We need to find the total displacement \(y_{total}\), which is the sum of the displacements of the two individual waves: $$y_{total} = y_1 + y_2$$

Use the trigonometric identity for the cosine of the sum of angles

Applying the identity \(\cos(A+B) = \cos A \cos B - \sin A \sin B\), we can rewrite the expression for \(y_{total}\) as: $$y_{total} = 2A\cos\left(\frac{\Delta \kappa x}{2}-\frac{\Delta \omega t}{2}\right)\cos\left(\!\left(\kappa x-\omega t\!\right)\!+\!\left(\!\frac{\Delta \kappa x}{2}-\frac{\Delta \omega t}{2}\right)\!\!\right)\!$$

Find the envelope function

To find the envelope function for the combined sine functions (i.e., the wave packet), we focus on the term that does not oscillate with respect to position and time: $$y_{env} = 2A\cos\left(\frac{\Delta \kappa x}{2}-\frac{\Delta \omega t}{2}\right)$$

Find the length of a wave packet and product

The length of a wave packet, \(\Delta x\), is the distance between two nodes of the envelope function, that is the distance between two points where the cosine term equals zero. To find this length, we should set the cosine term to zero: $$\frac{\Delta \kappa x}{2} = n\pi$$ $$\Delta x = \frac{2n\pi}{\Delta \kappa}$$ where \(n\) is an integer representing the \(n^{th}\) node. Now, we can find the product \(\Delta x \Delta \kappa\): $$\Delta x \Delta \kappa = 2n\pi$$ The value of the product \(\Delta x \Delta \kappa\) is \(2n\pi\), where \(n\) is an integer representing the \(n^{th}\) node.

Key concepts

sinusoidal waves

A sinusoidal wave is one of the most basic and important types of waveforms found in nature and engineering. It is described by its amplitude, frequency, and phase, and appears as a smooth, repetitive oscillation. Mathematically, a sinusoidal wave is typically represented by the function:

• The amplitude defines how tall or powerful the wave is.
• The frequency tells how often the wave peaks occur in a unit of time.
• The phase shifts the wave back and forth along the horizontal axis.

In physics, sinusoidal waves help us model a variety of phenomena, from sound waves to alternating current in electrical circuits. In the context of this exercise, we are looking at two sinusoidal waves combining together to produce what are known as wave packets. Wave packets result when these waves interfere, forming a pattern called beats, where the intensity and location of the peaks and troughs change over time.

wave number

The concept of wave number is critical in understanding wave behavior. It is defined as the number of wavelengths per unit distance and gives insight into the spatial frequency of the wave. The wave number is denoted by the Greek letter kappa (\(\kappa\)) and is given by the formula:\[k = \frac{2\pi}{\lambda}\]where \(\lambda\) is the wavelength.

• A higher wave number indicates a shorter wavelength and more cycles per unit distance.
• A lower wave number indicates a longer wavelength and fewer cycles.

In our problem, two waves with different wave numbers \(\kappa\) and \(\kappa + \Delta \kappa\) combine, leading to interesting effects such as the formation of a wave packet. Understanding the change \(\Delta \kappa\) between the wave numbers helps us determine properties like the length of a wave packet.

angular frequency

Angular frequency is another crucial parameter for describing oscillatory motion. It is denoted by the Greek letter omega (\(\omega\)), and it describes how quickly the wave oscillates in time. The relationship between angular frequency and the wave's period is:\[\omega = 2\pi f\]where \(f\) is the frequency.

• Angular frequency is measured in radians per second.
• A higher angular frequency means faster oscillations.
• A lower angular frequency means slower oscillations.

In combining the two sinusoidal waves in our exercise, their angular frequencies are \(\omega\) and \(\omega + \Delta \omega\). The difference \(\Delta \omega\) informs us on how the wave interference affects the resultant wave packet, influencing frequency-related behaviors such as beating.

trigonometric identity

Trigonometric identities are mathematical equations that relate the angles and sides of triangles. They are immensely useful in manipulating and simplifying expressions involving trigonometric functions. One such identity, known as the sum-to-product identity, plays a key role in the exercise:\[\cos(A + B) = \cos A \cos B - \sin A \sin B\]

• This identity allows us to take complex trigonometric expressions and break them into more manageable forms.
• It's crucial for transforming the addition of two cosine waves into a product form, revealing the underlying patterns.

When dealing with the interference of two sinusoidal waves, using this identity lets us find the envelope of the wave packet. This simplifies the computations and helps identify how the combined wave behaves, like generating beats and nodes.