Standing Wave Functions: Videos & Practice Problems

Standing waves result from the superposition of two waves traveling in opposite directions, characterized by a wave function of the form y(x,t)=Asin(kx)sin(ωt). The amplitude of the standing wave is double that of the individual waves. Key properties include the relationship between string length l and wavelength λ, given by l = n · λ/2, and the period of oscillation T = 2π/ω.

Properties of Standing Waves from Wave Functions

In the study of standing waves, understanding the wave function is crucial for calculating various properties such as the length of the string and the period of oscillation. The wave function for a standing wave can be expressed as:

y(x, t) = A_{sw} \cdot \sin(kx) \cdot \sin(\omega t)

Here, A_sw represents the amplitude of the standing wave, which is twice the amplitude of the individual waves that create it. The terms k and ω correspond to the wave number and angular frequency, respectively. The amplitude of the standing wave can be calculated as:

A = \frac{A_{sw}}{2}

For example, if the amplitude of the standing wave is 5 cm, the amplitude of the individual waves would be 2.5 cm. The number of loops in the standing wave is determined by the harmonic number n. For the third harmonic (n = 3), there are three loops, four nodes, and three antinodes.

To find the length of the string, we can use the relationship between the wavelength λ and the length of the string L given by:

\lambda_n = \frac{2L}{n}

Rearranging this gives:

L = \frac{n \cdot \lambda_n}{2}

To determine the wavelength for the third harmonic, we can relate the wave number k to the wavelength:

k = \frac{2\pi}{\lambda}

Thus, the wavelength can be calculated as:

\lambda = \frac{2\pi}{k}

For instance, if k is 0.34, the wavelength would be approximately 185 cm. Plugging this value back into the equation for L gives:

L = \frac{3 \cdot 185}{2} = 278 \text{ cm}

Finally, to calculate the period of oscillation T, we use the relationship:

\omega = \frac{2\pi}{T}

Rearranging this yields:

T = \frac{2\pi}{\omega}

If the angular frequency ω is given as 50 rad/s, the period can be calculated as:

T = \frac{2\pi}{50} \approx 1.3 \text{ seconds}

By understanding these relationships and calculations, one can effectively analyze the properties of standing waves and their behavior in various physical contexts.

Example 1

In the study of standing waves, understanding the positions of nodes and antinodes is crucial. A standing wave can be represented by a wave function, typically expressed as:

y(x, t) = A \sin(kx) \sin(\omega t)

Here, A is the amplitude, k is the wave number, and \omega is the angular frequency. The nodes are points along the wave where the displacement is always zero, regardless of time. To find the positions of these nodes, we focus on the condition where the wave function equals zero:

y(x, t) = 0

Since the sine function oscillates, we can simplify our analysis by setting:

\sin(kx) = 0

The sine function equals zero at integer multiples of π, which leads us to the general condition:

kx = n\pi

where n is an integer (0, 1, 2, ...). To find the corresponding values of x, we rearrange this equation:

x = \frac{n\pi}{k}

Substituting the wave number k into the equation allows us to calculate the specific positions of the nodes. For example, if k is given as 0.75π, the positions of the nodes can be calculated as:

x = \frac{n\pi}{0.75\pi} = \frac{n}{0.75} = \frac{4n}{3}

This results in node positions at:

x = 0, \frac{4}{3}, \frac{8}{3}, 4, ...

In practical terms, this means the first few nodes occur at approximately 0 m, 1.33 m, 2.66 m, and 4 m, continuing indefinitely. Understanding this derivation is essential for analyzing standing waves in various physical contexts, such as musical instruments or engineering applications.

Here’s what students ask on this topic:

What is the wave function for a standing wave?

The wave function for a standing wave is given by:

$y(x,t)=Asin\left(\mathrm{kx}\right)sin\left(\mathrm{\omega t}\right)$

Here, $A$ is the amplitude of the standing wave, $k$ is the wave number, and $\omega$ is the angular frequency. This function represents the superposition of two waves traveling in opposite directions, resulting in a pattern of nodes and antinodes.

How do you calculate the length of a string with a standing wave?

The length of a string with a standing wave can be calculated using the relationship between the string length $l$ and the wavelength $\lambda$:

$l=n·\frac{\mathrm{\lambda }}{2}$

Here, $n$ is the harmonic number. For example, if the string is vibrating in its 3rd harmonic (n=3), the length of the string is:

$l=3·\frac{\mathrm{\lambda }}{2}$

What is the relationship between the amplitude of a standing wave and the original waves?

The amplitude of a standing wave, denoted as $A$, is twice the amplitude of the individual waves that form it. If the amplitude of the original and reflected waves is $a$, then:

$A=2a$

This doubling occurs because the two waves interfere constructively at certain points, leading to an increase in amplitude.

How do you calculate the period of oscillation for a standing wave?

The period of oscillation $T$ for a standing wave can be calculated using the angular frequency $\omega$:

$\omega =\frac{2}{\pi }/T$

Rearranging this equation to solve for $T$ gives:

$T=\frac{2}{\pi }/\omega$

By substituting the value of $\omega$ from the wave function, you can determine the period of oscillation.

What are nodes and antinodes in a standing wave?

Nodes and antinodes are key features of standing waves. Nodes are points where the displacement of the wave is always zero, meaning there is no movement at these points. Antinodes are points where the displacement reaches its maximum value, meaning these points experience the greatest movement. In a standing wave, nodes and antinodes are evenly spaced along the medium. The number of nodes and antinodes depends on the harmonic number $n$. For example, in the 3rd harmonic, there are 4 nodes and 3 antinodes.