Question

Write the equation for a standing wave that has three antinodes of amplitude \(2.00 \mathrm{~cm}\) on a \(3.00-\mathrm{m}\) -long string that is fixed at both ends and vibrates 15.0 times a second. The time \(t=0\) is chosen to be an instant when the string is flat. If a wave pulse were propagated along this string, how fast would it travel?

Short answer

Question: Determine the equation for the given standing wave and calculate the wave speed. Given: - Amplitude: 2.00 cm - Number of antinodes: 3 - String length: 3.00 m - Frequency: 15.0 Hz Answer: The equation for the standing wave is \(y(x, t) = 0.0400 \sin(\pi x) \cos(30\pi t)\) and the wave speed is 30.0 m/s.

Step-by-step solution

General equation of a standing wave

The general equation of a standing wave is given by: \(y(x, t) = 2A\sin(kx)\cos(\omega t)\) where \(y(x, t)\) represents the displacement of the string at position \(x\) and time \(t\), \(A\) is the amplitude of the wave, \(k\) is the wave number, and \(\omega\) is the angular frequency. Since it's given that there are 3 antinodes of amplitude 2.00 cm, the string length is 3.00 m, and the string vibrates 15.0 times a second, let's use this information to find the wave parameters \(A\), \(k\), and \(\omega\).

Determine the wave parameters

First, we have the amplitude of each wave: \(A = 2.00 \mathrm{~cm} = 0.0200 \mathrm{~m}\) Next, we find the wavelength (\(\lambda\)) of the wave. Since the string has 3 antinodes and is fixed at both ends, it means that 1.5 wavelengths fit within the 3.00 m string length: \(1.5\lambda = 3.00 \mathrm{~m}\) Solving for \(\lambda\), we get: \(\lambda = 2.00 \mathrm{~m}\) Now we can find the wave number (\(k\)), which is related to the wavelength by the formula: \(k = \frac{2\pi}{\lambda}\) Substituting the values, we get: \(k = \frac{2\pi}{2.00 \mathrm{~m}} = \pi \mathrm{~m^{-1}}\) Lastly, we determine the angular frequency (\(\omega\)) using the given frequency (15.0 Hz): \(\omega = 2\pi f = 2\pi (15.0 \mathrm{~Hz}) = 30\pi \mathrm{~rad/s}\)

Write the equation for the standing wave

Now, we can plug the values of \(A\), \(k\), and \(\omega\) into the general equation for the standing wave: \(y(x, t) = 2(0.0200 \mathrm{~m})\sin(\pi x)\cos(30\pi t) = 0.0400 \mathrm{~m} \sin(\pi x) \cos(30\pi t)\) Thus, the equation for the standing wave is: \(y(x, t) = 0.0400 \sin(\pi x) \cos(30\pi t)\)

Calculate the wave speed

To find the speed (\(v\)) of the wave pulse along the string, we use the speed-frequency-wavelength relationship: \(v = f\lambda\) Substituting the frequency (15.0 Hz) and wavelength (2.00 m), we find the wave speed: \(v = (15.0 \mathrm{~Hz})(2.00 \mathrm{~m}) = 30.0 \mathrm{~m/s}\) So, the wave pulse would travel at a speed of 30.0 m/s along the string.

Key concepts

Wave Parameters

When studying waves, particularly standing waves, it's essential to understand the wave parameters that describe their physical characteristics.

The amplitude (\( A \)) of a wave is the maximum displacement of the wave from its rest position. It indicates the energy carried by the wave, with higher amplitudes carrying more energy. In the context of the standing wave on the string, the amplitude is given as 2.00 cm, which translates to the peak height of the wave's oscillation from its center position.

The wavelength (\( \text{λ} \text{(lambda)} \)), which is not directly provided in this example but can be calculated, represents the spatial period of the wave—the distance over which the wave's shape repeats. By using the information about the antinodes, we determined that 1.5 wavelengths fit within the 3.00 m string. This insight allows us to solve for the wave's actual wavelength.

Frequency (\( f \)) is the number of occurrences of a repeating event per unit time—in this case, how many cycles the wave completes in one second. The problem states a vibration frequency of 15.0 times per second (15.0 Hz), which directly influences the angular frequency and the wave speed.

Wave Number (k)

The wave number (\( k \)) is a critical concept in wave mechanics, particularly in the description of a standing wave. It is defined as the spatial frequency of a wave, representing the number of wave cycles that exist over a unit distance. The wave number is inversely proportional to the wavelength and is expressed by the relation \( k = \frac{2\pi}{\text{λ}} \).

In our example, we calculate the wave number using the wavelength of 2.00 m, resulting in a wave number of \( \pi \) m^-1. This value is crucial for defining the spatial variation of the standing wave along the length of the string and is utilized in the wave equation alongside the angular frequency.

Angular Frequency (ω)

Angular frequency (\( \text{ω} \text{(omega)} \)) plays a pivotal role in the temporal aspect of wave motion. Angular frequency relates to the standard frequency, which is the number of cycles per second, by the formula \( \text{ω} = 2\pi f \).

This parameter provides a measure of how quickly the wave oscillates in time. In this problem, with a frequency of 15.0 Hz, the angular frequency comes out to be \( 30\pi \) rad/s. The radian unit is used as it's a measure of angle, consistent with the wave's cyclic nature—how it repeats after a certain angle around a circle. The angular frequency is key to the time-dependent part of the standing wave equation, dictating how the wave oscillates with respect to time.

Wave Speed

Lastly, the wave speed (\( v \)) is a fundamental parameter that describes how fast a wave travels through a medium. For a standing wave on a string, wave speed is determined by the string's tension and mass per unit length, but it also follows the formula \( v = f\text{λ} \), which relates it directly to frequency and wavelength.

In the exercise, by multiplying the frequency (15.0 Hz) and the wavelength (2.00 m), we determined the wave speed to be 30.0 m/s. This speed indicates how fast a single point on the wave (like a pulse) would travel along the string. Understanding how to calculate wave speed from the frequency and wavelength is instrumental for many applications, ranging from musical instruments to modern communication technologies.