A standing wave on a 1.85-m-long horizontal string displays three loops when the string vibrates at 125 Hz. The maximum swing of the string (top to bottom) at the center of each loop is 8.00 cm. What is the function describing the standing wave?

Name: Physics for Scientists and Engineers
Author: Giancoli Douglas
ISBN: 9780137488179

Verified step by step guidance

Step 1: Understand the problem. A standing wave is formed on a string with three loops, meaning the string vibrates in its third harmonic. The string length is 1.85 m, the frequency is 125 Hz, and the amplitude (maximum displacement) is 8.00 cm. We need to derive the wave function describing this standing wave.

Step 2: Determine the wavelength of the standing wave. For the third harmonic, the string length corresponds to 1.5 wavelengths. Use the relationship \( L = \frac{n \lambda}{2} \), where \( L \) is the string length, \( n \) is the harmonic number (3 in this case), and \( \lambda \) is the wavelength. Solve for \( \lambda \).

Step 3: Calculate the wave speed. Use the formula \( v = f \lambda \), where \( v \) is the wave speed, \( f \) is the frequency (125 Hz), and \( \lambda \) is the wavelength determined in Step 2.

Step 4: Write the general form of the standing wave function. The standing wave can be described as \( y(x, t) = 2A \sin(kx) \cos(\omega t) \), where \( A \) is the amplitude (4.00 cm, since the maximum swing is top to bottom), \( k \) is the wave number \( k = \frac{2\pi}{\lambda} \), and \( \omega \) is the angular frequency \( \omega = 2\pi f \). Substitute the known values for \( A \), \( k \), and \( \omega \).

Step 5: Finalize the wave function. Combine the values from previous steps into the equation \( y(x, t) = 2A \sin(kx) \cos(\omega t) \), ensuring all units are consistent (meters for length, seconds for time). This function describes the standing wave on the string.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. This results in a wave pattern that appears to be stationary, characterized by nodes (points of no displacement) and antinodes (points of maximum displacement). In this case, the three loops indicate the presence of three antinodes along the string.

Wave Function

The wave function describes the displacement of points on a wave as a function of time and position. For standing waves, this can often be expressed in the form of a sine or cosine function, incorporating parameters such as amplitude, wavelength, and frequency. The general form for a standing wave can be written as y(x, t) = A sin(kx) cos(ωt), where A is the amplitude, k is the wave number, and ω is the angular frequency.

Frequency and Wavelength

Frequency is the number of oscillations or cycles that occur in a unit of time, measured in Hertz (Hz). Wavelength is the distance between successive points of the same phase on a wave, such as crest to crest. The relationship between frequency (f), wavelength (λ), and wave speed (v) is given by the equation v = fλ, which is essential for determining the characteristics of the standing wave in the string.

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