Question

Vibrations from a \(622-\mathrm{Hz}\) tuning fork set up standing waves in a string clamped at both ends. The wave speed for the string is \(388 \mathrm{~m} / \mathrm{s}\). The standing wave has four loops and an amplitude of \(1.90 \mathrm{~mm} .\) (a) What is the length of the string? (b) Write an equation for the displacement of the string as a function of position and time.

Short answer

The length of the string is \(2 \lambda\) and the displacement of the string as a function of position and time is \(y(x, t) = A \sin(kx - \omega t)\).

Step-by-step solution

Calculate the Wavelength

To calculate the wavelength \(\(\lambda\)\) of the standing waves use the formula for speed \(v = f \lambda\), where \(f = 622 Hz\) is the frequency and \(v = 388 m/s\) is the speed of the wave. Solving for \(\lambda\) yields \(\lambda = \(\frac{v}{f}\) = \(\frac{388 m/s}{622 Hz}\)\).

Calculate the Length of the String

Knowing that the standing wave has four loops and each loop is equivalent to half a wavelength, the length of the string is \(2 \lambda\). So, multiply the discovered wavelength by 2 to get the length of the string: \(Length = 2 \lambda\).

Write the Wave Function

The vertical position of a point on a standing wave is described by the wave function \(y(x, t) = A \sin(kx - \omega t)\), where \(x\) is the position along the string, \(t\) is time, \(A = 1.90 mm\) is the amplitude of the wave, \(k = \(\frac{2 \pi}{\lambda}\)\) is the wave number, and \(\omega = 2 \pi f\) is the angular frequency. Substitute the given values into the equation to define the displacement of the string over time.

Key concepts

Wave Speed

Wave speed, denoted as 'v,' is a fundamental property of waves that represents how fast a wave propagates through a medium. It is dependent on the type of wave and the properties of the medium through which it travels. For example, sound waves travel faster in solids compared to liquids and gases because particles in solids are more closely packed together.

In the context of our exercise, the wave speed is given as 388 m/s for a string on which standing waves are created. It's important to note that this speed is the result of the tension and mass per unit length of the string; it's a property intrinsic to the string and independent of the frequency of the tuning fork used to generate the waves.

Wavelength Calculation

Wavelength, commonly represented by the Greek letter lambda \(\lambda\), is the distance between two consecutive points that are in phase on a wave, such as two crests or troughs. The calculation of wavelength is crucial in understanding wave properties and behaviors.

To compute the wavelength of a standing wave, you can use the formula \(v = f\lambda\), where 'v' stands for wave speed and 'f' for the frequency of the vibrations producing the wave. In our exercise, by rearranging the formula to \(\lambda = \frac{v}{f}\), and using the given values, the wavelength can be calculated. Knowing the wavelength then allows us to determine important characteristics of the wave, such as its standing wave patterns, which in turn leads to understanding the physical dimensions that support these patterns, like the length of the string in our example.

Wave Function

A wave function is a mathematical description of a wave's displacement over time and position. It is particularly important in understanding and predicting the behavior of waves in various physical contexts. For standing waves, the wave function provides a way to visualize how the wave oscillates in space and time.

The standard form of the wave function for a simple harmonic wave on a string is \(y(x, t) = A \sin(kx - \omega t)\), where 'y' is the vertical displacement, 'x' is the position along the string, 't' is time, 'A' is the amplitude, 'k' is the wave number, and '\omega' is the angular frequency. The wave number and angular frequency are related to the wavelength and frequency, providing a direct link between the physical and mathematical representations of the wave. By substituting the wavelength and frequency values into this function, as it was done in our example, anyone can describe the exact motion of every point on the string at any given time.