Question

A sinusoidal transverse wave of wavelength \(20.0 \mathrm{~cm}\) and frequency 500\. Hz travels along a string in the positive \(z\) -direction. The wave oscillations take place in the \(x z\) -plane and have an amplitude of \(3.00 \mathrm{~cm}\). At time \(t=0,\) the displacement of the string at \(x=0\) is \(z=3.00 \mathrm{~cm}\) a) A photo of the wave is taken at \(t=0 .\) Make a simple sketch (including axes) of the string at this time. b) Determine the speed of the wave. c) Determine the wave's wave number. d) If the linear mass density of the string is \(30.0 \mathrm{~g} / \mathrm{m},\) what is the tension in the string? e) Determine the function \(D(z, t)\) that describes the displacement \(x\) that is produced in the string by this wave.

Short answer

Answer: The displacement function for the given wave is D(z, t) = 0.03sin(31.4z - 1000πt + π/2).

Step-by-step solution

a) Sketch of the wave at t=0

At t=0, the displacement of the string at x=0 is z=3 cm, which means that the string is at its highest point since the amplitude of the wave is 3 cm. We have to sketch a sinusoidal wave in the xz-plane with its highest point at the origin (0, 3 cm) and wavelength 20 cm. Remember that the wave is moving in the positive z-direction, and oscillations are taking place in the xz-plane.

b) Determine the speed of the wave

To calculate the speed of the wave, we will use the formula: speed = frequency x wavelength v = fλ \\ v = (500 \ \mathrm{Hz})(0.2 \ \mathrm{m}) \\ v = 100 \ \mathrm{m/s} \\ So, the speed of the wave is 100 m/s.

c) Determine the wave's wave number

To calculate the wave number (k) we will use the formula: k = 2π/λ \\ k = \frac{2 \pi}{0.2 \ \mathrm{m}} \\ k \approx 31.4 \ \mathrm{m^{-1}} \\ The wave's wave number is approximately 31.4 m^{-1}.

d) Calculate the tension in the string

To find the tension in the string, we will use the formula: tension = linear density x speed^2 T = μv^2 \\ T = \left(\frac{30.0 \ \mathrm{g}}{\mathrm{m}} \times \frac{1 \ \mathrm{kg}}{1000\ \mathrm{g}}\right)(100 \ \mathrm{m/s})^2 \\ T \approx 300 \ \mathrm{N} \\ The tension in the string is approximately 300 N.

e) Determine the function D(z, t) that describes the displacement x.

We will use the general equation for a sinusoidal wave moving in the positive z-direction: D(z, t) = A sin(kz - ωt + φ) \\ Here, A is the amplitude (3 cm = 0.03 m), k is the wave number (31.4 m^{-1}), and ω is the angular frequency (2πf = 2π(500 Hz) = 1000π rad/s). At t=0, x=0 and z=3 cm = 0.03 m, which gives us: 0 = 0.03 \sin(0 - 0 + φ) \\ 0.03\ \mathrm{m} = 0.03 \sin φ \\ φ = \pi/2 \\ Now, we can put these values back into the general equation to obtain the displacement function: D(z, t) = 0.03 \sin(31.4z - 1000πt + π/2) \\ So, the displacement function D(z, t) is: D(z, t) = 0.03sin(31.4z - 1000πt + π/2).

Key concepts

Wavelength

Wavelength (λ) is a key characteristic of a wave and is the distance between two consecutive points that are in phase. In simple terms, it's the length of one complete wave cycle, such as the distance from crest to crest or trough to trough in a transverse wave like the one described in the exercise. Wavelength is measured in meters (m) and directly affects the wave's speed and energy. Understanding wavelength is crucial for solving problems involving wave speed and frequency, as it's part of the fundamental relationship v = fλwhere v is the wave speed and f is the frequency.

Frequency

Frequency (f) of a wave refers to how many complete wave cycles pass a given point per second. It is measured in hertz (Hz), which is equivalent to 1/s. In our exercise, the frequency of the transverse wave is given as 500 Hz, meaning that 500 wavelengths pass a given point each second. The frequency is an essential component when determining other wave properties such as speed and energy, and it is inversely proportional to the wavelength, maintaining the relationship that higher frequencies result in shorter wavelengths and vice versa.

Wave Speed

Wave speed (v) is the rate at which the wave propagates through a medium. In the context of a transverse wave on a string, as in the exercise, the wave speed can be determined by multiplying its frequency (f) by its wavelength (λ), which is expressed by the equation v = fλ. For the wave in the exercise, the speed is calculated to be 100 m/s, indicating how fast the wave's energy and information are traveling along the string.

Wave Number

Wave number (k), not to be confused with wavelength, is a spatial frequency that describes the number of wave cycles in a unit distance. It's often used in the study of wave phenomena in physics and engineering. The formula to calculate wave number is given by k = 2π/λ, where λ is the wavelength. In our example, the computed wave number is approximately 31.4 m^-1, which tells us how many cycles of the wave are present per meter of the wave's propagation along the z-axis.

Linear Mass Density

Linear mass density (μ) represents the mass per unit length of the string on which the wave is traveling and is usually expressed in kilograms per meter (kg/m). It is a critical factor when it comes to calculating the tension required to produce a certain wave speed in a string. The higher the linear mass density, the greater the tension needed to maintain the same wave speed. In the given problem, a linear mass density of 30.0 g/m translates to 0.03 kg/m, which was used to calculate the tension in the string responsible for the wave's propagation.

Tension in a String

Tension in a string is the force exerted along the string's length, and it plays a vital role in determining the wave speed on the string. For waves on a string, tension can be calculated using the equation T = μv^2 where T is the tension, μ is the linear mass density, and v is the wave speed. The exercise shows a practical application of this principle, where a tension of approximately 300 N is determined necessary to sustain a wave speed of 100 m/s on a string with the given mass density.

Sinusoidal Wave Equation

The sinusoidal wave equation is a mathematical description of a wave's displacement over time and position. It's given by the general form D(z, t) = A sin(kz - ωt + φ), where D(z, t) describes the displacement, A is the amplitude, k is the wave number, ω is the angular frequency, t is time, φ is the phase constant, and z is the position along the direction of wave propagation. This equation reflects the harmonic nature of sinusoidal waves and incorporates wave parameters like amplitude and frequency to detail how the wave evolves spatially and temporarily.

Using the given values in the exercise, including the determined phase constant of π/2, we derived a specific sinusoidal wave equation describing the transverse wave's motion on the string.