Question

A sound wave in a fluid medium is reflected at a barrier so that a standing wave is formed. The distance between nodes is \(3.84 \mathrm{~cm}\) and the speed of propagation is \(1520 \mathrm{~m} / \mathrm{s}\). Find the frequency.

Short answer

The frequency of the standing wave in the fluid is approximately \(19805 Hz\).

Step-by-step solution

Convert the Distance between Nodes into Wavelength

The distance between nodes in the standing wave is given as \(3.84 cm\). However, this represents half a wavelength. So, the total wavelength \(\lambda\) is \(2 \times 3.84 cm = 7.68 cm\). Convert it to meters by dividing by 100, so \(\lambda = 0.0768~m\).

Calculate the Frequency

The speed of sound \(v\) in the medium is \(1520 m/s\), and it equals the product of the wavelength \(\lambda\) and the frequency \(f\), i.e., \(v = f \cdot \lambda\). Substituting the given values we can find the frequency \(f\) by re-arranging to \(f = v / \lambda\). This gives \(f = 1520~m/s / 0.0768~m = 19805~Hz\).

Key concepts

Understanding Standing Waves

Standing waves are a unique type of wave that occurs when two waves of the same frequency and amplitude travel in opposite directions through a medium and interfere with each other. These waves are characterized by their stationary pattern, consisting of nodes and antinodes.

- **Nodes:** Points along the medium where no movement occurs. They are formed because the interference between the two waves results in destructive interference at these locations.

- **Antinodes:** Points where the maximum displacement occurs. These are located halfway between nodes and result from constructive interference.

In the context given, the distance between nodes is crucial for calculating the wavelength. Remember, this distance represents half of the wavelength. By recognizing this fact, you can find the total wavelength essential for other calculations.

Measuring the Speed of Sound

The speed of sound in a particular medium determines how fast sound waves travel through it. In general, sound speed depends on several factors, including:

- **Medium type:** Sound travels faster through solids, then liquids, and slowest through gases.

- **Temperature:** Warmer temperatures generally increase sound speed because the molecules move more quickly.

Knowing that the speed of sound in our problem is given as \(1520~m/s\), this implies we're likely dealing with a liquid medium. In practice, using the speed of sound is vital for calculating other properties, such as wavelength and frequency of the standing wave.

Wavelength Calculation and Its Role

Wavelength is a fundamental measurement in the study of waves, defined as the distance between two consecutive points in phase on a wave, such as crest to crest or trough to trough.

In the context of standing waves in the exercise, the wavelength \(\lambda\) plays a crucial role in linking speed and frequency through the wave equation: \(v = f \cdot \lambda\). By knowing any two of these properties, you can compute the third.

For example, in the exercise provided, the halved distance between nodes is initially \(3.84~cm\). Converting this to full wavelength involves doubling this value, resulting in \(7.68~cm\) or \(0.0768~m\) in meters. This wavelength then connects with the wave speed to find frequency: \(f = v / \lambda = 1520~m/s / 0.0768~m\), leading to a calculated frequency of \(19805~Hz\).