Phase Constant: Videos & Practice Problems

Wave functions can be expressed using sine or cosine functions, depending on their starting positions. For a wave with an amplitude of 4 meters, oscillating between 4 and -4, starting at a displacement of 3, a phase constant (φ) is introduced to account for shifts. The wave function can be written as $y^{x,t}=4\sin (10x-62.8t+0.85)$ or $y^{x,t}=4\cos (10x-62.8t-0.72)$.

Phase Constant of a Wave Function

In wave mechanics, wave functions can be represented using sine or cosine functions, depending on their starting positions. Typically, waves start at either the equilibrium position (y = 0) or at their maximum amplitude. However, there are instances where a wave begins at a different displacement, necessitating the use of a phase constant to accurately describe its behavior.

Consider a wave with an amplitude of 4 meters, oscillating between 4 and -4 meters, starting at a displacement of 3 meters when both time (t) and position (x) are zero. To write the wave function for this scenario, we can utilize the sine function, which is generally expressed as:

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

Here, A represents the amplitude, k is the wave number, ω is the angular frequency, and φ is the phase constant. The phase constant allows us to shift the wave function horizontally, aligning it with the actual starting position of the wave.

For this specific wave, the wave number k is given as 10 radians per meter, and the angular frequency ω is 62.8 radians per second. Since the wave starts at a displacement of 3 meters, we need to determine the phase constant φ that corresponds to this initial condition.

To find φ, we first set up the equation:

4 \sin(10(0) - 62.8(0) + \phi) = 3

This simplifies to:

4 \sin(\phi) = 3

From this, we can isolate the sine function:

\sin(\phi) = \frac{3}{4}

Taking the inverse sine gives us:

\phi = \arcsin\left(\frac{3}{4}\right) \approx 0.85 \text{ radians}

Thus, the complete wave function using the sine function is:

y(x, t) = 4 \sin(10x - 62.8t + 0.85)

Alternatively, we can express the same wave using a cosine function. The general form for the cosine wave function is:

y(x, t) = A \cos(kx - \omega t + \phi_c)

In this case, since the wave starts at a displacement of 3 meters, we find that the phase constant for the cosine function will be negative, indicating a shift to the right. Following similar steps as before, we set up the equation:

4 \cos(10(0) - 62.8(0) + \phi_c) = 3

This simplifies to:

4 \cos(\phi_c) = 3

Isolating the cosine function gives:

\cos(\phi_c) = \frac{3}{4}

Taking the inverse cosine yields:

\phi_c = \arccos\left(\frac{3}{4}\right) \approx 0.72 \text{ radians}

Thus, the complete wave function using the cosine function is:

y(x, t) = 4 \cos(10x - 62.8t - 0.72)

In summary, both sine and cosine functions can effectively describe the same wave, but the phase constants differ due to the inherent properties of these functions. Understanding how to manipulate these constants is crucial for accurately modeling wave behavior in various contexts.

A wave traveling to the right has an Amplitude of 15 cm, wavelength of 40 cm, and oscillates 8 times per second. At t = 0, the displacement of a particle at x = 0 along this wave is +15 cm. Write the wave function, including the phase constant, using a sine function.

• A. $y\left(x,t\right)=0.15\sin\left(0.40x-8t-1.57\right)$
• B. $y\left(x,t\right)=0.15\sin\left(15.7x+50.3t\right)$
• C. $y\left(x,t\right)=0.15\sin\left(0.157x-0.78t-1.57\right)$
• D. $y\left(x,t\right)=0.15\sin\left(15.7x-50.3t+1.57\right)$

Answer: (D)

Here’s what students ask on this topic:

What is a phase constant in wave functions?

A phase constant, denoted as φ (phi), is a term added to the argument of sine or cosine functions in wave equations to account for shifts in the wave's starting position. It essentially shifts the wave left or right from its normal starting point. For example, in the wave function y(x,t) = A sin(kx - ωt + φ), the phase constant φ adjusts the wave's phase, allowing it to start at a position other than y = 0 or the amplitude. This is crucial for accurately describing waves that do not start at these special points.

How do you determine the sign of the phase constant?

The sign of the phase constant φ depends on the direction of the shift from the wave's normal starting position. If the wave is shifted to the right, the phase constant is negative. Conversely, if the wave is shifted to the left, the phase constant is positive. This rule is similar to determining the sign of the terms in the wave function's argument, such as kx - ωt. For example, if a wave is described by y(x,t) = A sin(kx - ωt + φ) and is shifted to the left, φ will be positive.

How do you calculate the phase constant in a wave function?

To calculate the phase constant φ, follow these steps: 1) Write the wave function including φ, e.g., y(x,t) = A sin(kx - ωt + φ). 2) Determine the sign of φ based on the wave's shift direction. 3) Plug in the given values for amplitude, wave number (k), angular frequency (ω), and initial displacement. 4) Solve for φ using the initial conditions. For instance, if y(0,0) = 3 and A = 4, solve 4 sin(φ) = 3 to find φ = arcsin(3/4).

Can you use both sine and cosine functions to describe the same wave?

Yes, you can use both sine and cosine functions to describe the same wave. The choice between sine and cosine depends on the wave's starting position. For example, a wave starting at y = 0 is typically described using a sine function, while a wave starting at its amplitude is described using a cosine function. However, by introducing a phase constant φ, you can shift either function to match the wave's actual starting position. For instance, y(x,t) = A sin(kx - ωt + φ) and y(x,t) = A cos(kx - ωt - φ) can describe the same wave with different phase constants.

What is the significance of the phase constant in wave equations?

The phase constant φ in wave equations is significant because it allows for the accurate representation of waves that do not start at standard positions (y = 0 or amplitude). It shifts the wave function horizontally, ensuring that the wave's initial conditions are met. This is crucial for solving real-world problems where waves often start at arbitrary positions. For example, in the wave function y(x,t) = A sin(kx - ωt + φ), the phase constant φ ensures that the wave starts at the correct displacement, making the mathematical model align with physical observations.