Wave Functions: Videos & Practice Problems

The wave function, represented as $y(x,t)$, describes oscillating waves using sine or cosine functions. Key variables include amplitude $a$, wave number $k$ (where $k=\frac{2}{\lambda }$), and angular frequency $\omega$ (where $\omega =\frac{2}{\pi }$). The wave speed $v$ can be calculated using $v=\frac{\omega }{k}$. Understanding the distinction between propagation velocity and transverse velocity is crucial for analyzing wave behavior.

Intro to Wave Functions

The wave function is a crucial concept in understanding wave behavior, particularly in physics. It is represented by a sinusoidal equation, which can be either a sine or cosine function, depending on the initial conditions of the wave. The general form of the wave function is given by:

$$y(x, t) = A \cdot \cos(kx \pm \omega t)$$

or

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

Here, \(y\) represents the displacement of a particle in the wave at a given position \(x\) and time \(t\). The choice between sine and cosine depends on the starting point of the wave: if the wave starts at the maximum amplitude, the cosine function is used; if it starts at zero displacement, the sine function is appropriate.

In this equation, three key variables are essential:

• A (Amplitude): This is the maximum displacement from the equilibrium position, indicating how far the wave oscillates from its rest position.
• k (Wave Number): This is defined as \(k = \frac{2\pi}{\lambda}\), where \(\lambda\) is the wavelength. The units of \(k\) are radians per meter.
• \(\omega\) (Angular Frequency): This is given by \(\omega = 2\pi f\) or \(\omega = \frac{2\pi}{T}\), where \(f\) is the frequency and \(T\) is the period of the wave. The units of \(\omega\) are radians per second.

To determine the wave function for a specific scenario, one must first identify the amplitude, wave speed, and wavelength. For example, if the amplitude \(A\) is 0.5 meters, the wave speed \(v\) is 8 meters per second, and the wavelength \(\lambda\) is 0.32 meters, we can calculate the wave number and angular frequency:

1. Calculate \(k\):

$$k = \frac{2\pi}{\lambda} = \frac{2\pi}{0.32} \approx 19.6 \, \text{radians/m}$$

2. Calculate the frequency \(f\) using the wave speed equation \(v = \lambda f\):

$$f = \frac{v}{\lambda} = \frac{8}{0.32} \approx 25 \, \text{Hz}$$

3. Calculate \(\omega\):

$$\omega = 2\pi f = 2\pi \times 25 \approx 157.1 \, \text{radians/s}$$

With these values, the wave function can be expressed as:

$$y(x, t) = 0.5 \cdot \cos(19.6x - 157.1t)$$

To evaluate the displacement of a particle at a specific position and time, such as \(x = 0.4\) meters and \(t = 0.75\) seconds, substitute these values into the wave function:

$$y(0.4, 0.75) = 0.5 \cdot \cos(19.6 \cdot 0.4 - 157.1 \cdot 0.75)$$

Ensure that the calculator is set to radians mode for accurate results. After performing the calculation, the displacement is found to be approximately -0.5 meters, indicating that the particle is at the maximum downward displacement at that position and time.

Understanding the wave function and its components is essential for analyzing wave behavior in various physical contexts, from sound waves to electromagnetic waves.

A transverse harmonic wave moving to the left has a wavelength of 2.5 m and a wave speed of 12 m/s. The amplitude of the wave is 0.1 m. At x = 0 and t = 0, the displacement of the wave is y = 0. Write the wave function for this wave.

• A. $y\left(x,t\right)=0.1\sin\left(2.51x+30.2t\right)$
• B. $y\left(x,t\right)=0.1\cos\left(2.51x-30.2t\right)$
• C. $y\left(x,t\right)=0.1\sin\left(2.5x+15.7t\right)$
• D. $y\left(x,t\right)=0.1\sin\left(15.7x+188.5t\right)$

Answer: (A)

Example 1

In this discussion, we analyze a wave function represented as \( y(x, t) = 6.5 \cos\left(\frac{2\pi}{28} x + \frac{2\pi}{0.36} t\right) \). Our goal is to determine the wavelength, frequency, and direction of wave propagation.

To find the wavelength, we utilize the wave number \( k \), which is defined as \( k = \frac{2\pi}{\lambda} \). In our wave function, \( k \) is given as \( \frac{2\pi}{28} \). By rearranging the equation, we can solve for the wavelength \( \lambda \):

\[ \lambda = \frac{2\pi}{k} = 28 \text{ meters} \]

Next, we calculate the frequency \( f \) using the angular frequency \( \omega \), which is related to frequency by the equation \( \omega = 2\pi f \). In our case, \( \omega \) is given as \( \frac{2\pi}{0.36} \). To find \( f \), we rearrange the equation:

\[ f = \frac{\omega}{2\pi} = \frac{1}{0.36} \approx 2.78 \text{ hertz} \]

Finally, we determine the direction of wave travel. The general form of the wave function can be expressed as \( kx \pm \omega t \). The presence of a plus sign in the term \( kx + \omega t \) indicates that the wave travels to the left. Therefore, the wave propagates in the leftward direction.

In summary, we have established that the wavelength of the wave is 28 meters, the frequency is approximately 2.78 hertz, and the wave travels to the left.

Calculating Wave Speed from Wave Functions

In the study of wave mechanics, understanding the wave function is crucial for calculating wave speed. The wave function can be expressed as either a sine or cosine function, typically represented as \( y(x, t) \). To determine the wave speed \( v \), there are several equations available, but the choice of equation often depends on the information provided in the problem.

For transverse waves traveling along a string, one common equation is:

\( v = \sqrt{\frac{T}{\mu}} \)

where \( T \) is the tension in the string and \( \mu \) is the mass per unit length. However, if the tension or mass is not given, alternative methods must be employed.

Another fundamental relationship for all types of waves is:

\( v = \lambda f \)

where \( \lambda \) is the wavelength and \( f \) is the frequency. If these values are not provided, we can utilize the wave function parameters directly.

The wave number \( k \) and angular frequency \( \omega \) are related to wavelength and frequency, respectively. Specifically, the relationships are:

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

and

\( \omega = 2\pi f \).

By rearranging these equations, we can express wavelength and frequency in terms of \( k \) and \( \omega \):

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

and

\( f = \frac{\omega}{2\pi} \).

Substituting these into the wave speed equation gives:

\( v = \lambda f = \left(\frac{2\pi}{k}\right) \left(\frac{\omega}{2\pi}\right) = \frac{\omega}{k}.

This shortcut allows for the direct calculation of wave speed using the angular frequency \( \omega \) and wave number \( k \). For example, if \( \omega = 6 \, \text{rad/s} \) and \( k = 0.4 \, \text{m}^{-1} \), substituting these values yields:

\( v = \frac{6}{0.4} = 15 \, \text{m/s}.

Thus, the wave speed, also referred to as the propagation velocity, can be efficiently calculated using the wave function parameters without needing to derive wavelength or frequency separately. This method streamlines the process and reinforces the interconnectedness of wave properties.

Example 2

In this discussion, we explore the behavior of a wave function represented as \( y(x, t) = 6 \, \text{mm} \sin(kx - \omega t) \), where the amplitude is 6 millimeters, \( k \) is the wave number in radians per millimeter, and \( \omega \) is the angular frequency in radians per second. The amplitude indicates the maximum displacement of particles in the medium, which oscillate between +6 mm and -6 mm, creating a sine wave pattern.

To determine the time it takes for a particle on the string to travel from +6 mm to -6 mm, we first calculate the total displacement. This displacement is the distance between the crest and trough of the wave, which is twice the amplitude: \( \Delta x = 2 \times \text{amplitude} = 2 \times 6 \, \text{mm} = 12 \, \text{mm} \).

Next, we need to find the velocity of the particles moving up and down. The transverse velocity \( v \) can be calculated using the formula:

\[ v = \frac{\omega}{k} \]

Substituting the given values, where \( \omega = 600 \, \text{radians/second} \) and \( k = 5 \, \text{radians/mm} \), we have:

\[ v = \frac{600 \, \text{radians/second}}{5 \, \text{radians/mm}} = 120 \, \text{mm/second} \]

Now, we can find the time \( \Delta t \) it takes for the particle to complete the displacement using the rearranged formula:

\[ \Delta t = \frac{\Delta x}{v} \]

Substituting the values we calculated:

\[ \Delta t = \frac{12 \, \text{mm}}{120 \, \text{mm/second}} = 0.1 \, \text{seconds} \]

This result indicates that it takes 0.1 seconds for a particle on the string to oscillate from +6 mm to -6 mm. Understanding these relationships between amplitude, velocity, and time is crucial in analyzing wave motion and the behavior of particles in a medium.

Transverse Velocity of Waves

In wave mechanics, understanding the different types of velocities associated with waves is crucial. Two key concepts are the propagation velocity and the transverse velocity. The propagation velocity, also known as the wave velocity, refers to the speed at which the overall wave pattern travels through space. This can be calculated using the formula:

$$v = \frac{\omega}{k}$$

where \( \omega \) is the angular frequency and \( k \) is the wave number. For example, if \( \omega = 6 \, \text{rad/s} \) and \( k = 0.4 \, \text{rad/m} \), the propagation velocity would be:

$$v = \frac{6}{0.4} = 15 \, \text{m/s}$$

This indicates that the wave pattern moves to the right at 15 meters per second.

On the other hand, the transverse velocity describes the motion of particles within the wave as they oscillate perpendicular to the direction of wave propagation. This velocity can be calculated using different equations depending on whether the wave function is expressed in terms of sine or cosine. For a wave function of the form \( y(x, t) = a \cos(kx - \omega t) \), the transverse velocity is given by:

$$v_{t} = \omega a \sin(kx - \omega t)$$

Conversely, if the wave function is expressed as \( y(x, t) = a \sin(kx - \omega t) \), the equation becomes:

$$v_{t} = \pm \omega a \cos(kx - \omega t)$$

Here, the sign depends on the sign in the wave function. For instance, if the wave function has a minus sign, the transverse velocity will have a plus sign, and vice versa.

To calculate the transverse velocity at a specific position and time, substitute the values into the appropriate equation. For example, if \( a = 3 \), \( \omega = 6 \), \( k = 0.4 \), \( x = 0.75 \), and \( t = 0.2 \), the transverse velocity can be computed as:

$$v_{t} = 18 \sin(0.4 \times 0.75 - 6 \times 0.2)$$

Calculating this yields a transverse velocity of approximately -14.1 m/s, indicating that the particle is moving downwards at that moment.

It's important to note that while the propagation velocity remains constant across the wave, the transverse velocity varies with both position and time, meaning different particles in the wave can move in different directions at any given moment.

Finally, the maximum transverse velocity can be determined using the formula:

$$v_{t \, \text{max}} = \omega a$$

For the given values, this results in:

$$v_{t \, \text{max}} = 6 \times 3 = 18 \, \text{m/s}$$

This maximum value represents the highest speed at which any particle in the wave can oscillate up and down.

Here’s what students ask on this topic:

What is the wave function in physics?

The wave function in physics, denoted as y(x,t), is a mathematical description of an oscillating wave. It is typically represented using sinusoidal functions, either sine or cosine. The wave function provides the displacement of a particle on the wave at a given position (x) and time (t). The general form of the wave function is y(x,t) = A sin(kx ± ωt) or y(x,t) = A cos(kx ± ωt), where A is the amplitude, k is the wave number (k = 2π/λ), and ω is the angular frequency (ω = 2π/T). The wave function is crucial for understanding wave behavior and solving wave-related problems in physics.

How do you calculate the wave speed using the wave function?

The wave speed (v) can be calculated using the wave function by relating the angular frequency (ω) and the wave number (k). The formula for wave speed is given by:

$v=\frac{\omega }{k}$

Here, ω is the angular frequency, and k is the wave number. The wave number k is related to the wavelength (λ) by the equation k = 2π/λ, and the angular frequency ω is related to the period (T) by ω = 2π/T. By substituting these values into the formula, you can determine the wave speed.

What is the difference between propagation velocity and transverse velocity in wave functions?

Propagation velocity and transverse velocity are two different concepts in wave functions. Propagation velocity, also known as wave speed, refers to the speed at which the overall wave pattern moves through the medium. It is given by the formula:

$v=\frac{\omega }{k}$

Transverse velocity, on the other hand, refers to the velocity of individual particles in the medium as they move perpendicular to the direction of wave propagation. It is given by the derivative of the wave function with respect to time. The transverse velocity varies with position (x) and time (t), while the propagation velocity is constant for a given wave.

How do you determine whether to use sine or cosine in the wave function?

The choice between sine and cosine in the wave function depends on the initial conditions of the wave. If the wave starts at zero displacement (y = 0) at t = 0, you use the sine function. If the wave starts at its maximum displacement (positive or negative amplitude) at t = 0, you use the cosine function. Both sine and cosine functions are sinusoidal and describe the same wave behavior, but they differ in their phase. The general forms are:

$y(x,t)=Asin(\mathrm{kx}\pm \mathrm{\omega t})$

or

$y(x,t)=Acos(\mathrm{kx}\pm \mathrm{\omega t})$

What are the key variables in the wave function equation?

The key variables in the wave function equation y(x,t) = A sin(kx ± ωt) or y(x,t) = A cos(kx ± ωt) are:

• Amplitude (A): The maximum displacement of the wave from its equilibrium position.
• Wave number (k): Defined as k = 2π/λ, where λ is the wavelength. It represents the number of wave cycles per unit distance.
• Angular frequency (ω): Defined as ω = 2π/T, where T is the period. It represents the rate of oscillation of the wave.
• Position (x): The location along the wave where the displacement is being measured.
• Time (t): The moment at which the displacement is being measured.

These variables are essential for describing the behavior and characteristics of oscillating waves.