Phase Constant - Online Tutor, Practice Problems & Exam Prep

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

Guys, up until now, all of the wave functions waves that we've seen so far have either started at \( y = 0 \) like these graphs over here, or they started at the amplitudes like these graphs over here. Now occasionally, you're going to run into a situation or run into a problem where you'll have a graph that doesn't start at either one of those special points. The example we’re going to work out down here, we have a wave with an amplitude of 4 meters. It oscillates between 4 and negative 4, but the starting position at \( t = 0 \) and \( x = 0 \) is actually a displacement of 3. It starts here, not at 0 and not at the amplitudes. In this problem, we’re going to have to write a wave function for this wave. I'm going to show you how to write these wave functions by using something called the phase constants, and in textbooks this can be pretty confusing, but I'm going to show you a step-by-step process for how to deal with these kinds of problems. Let's go ahead and check this out here. So remember that we could use either sines or cosines to describe these graphs here, and it really just depended on where our starting locations were. If we started at \( y = 0 \), we used the sine, and if we started at \( y \) equals either one of the amplitudes, then we used the cosine graph. So then how do we use these, or how do we write a wave function for something that doesn't start at those sorts of special positions? It turns out that we can actually write any wave function by using either sine or cosine. We can actually describe this graph here by using sines or cosines. All we have to do is we just have to add a constant inside our parentheses here. So we have a sine or a cosine (eg. \( \sin(kx \pm \omega t \pm \phi) \)). And this \( \phi \) here represents or is called a phase constant, and what this basically does is it shifts the graph to the left or to the right from a normal starting point.

So basically, the only difference between these graphs here is where they start. So we can take this graph and we can actually shift it to the left or to the right, and we’d be able to sort of line it up and make it look like these other graphs over here or basically any point in between. And that’s what that phase constant does as it shifts. I’m going to get back to this in just a second here. We're told that the characteristics of the properties of this wave, the wave number is 10 radians per meter. \( k = 10 \). We have the angular frequency, \( \omega = 62.8 \), and we also have the amplitude which is 4. Here are all our values. We want to write the wave function using a sine function. Remember, we can use because these graphs are sinusoidal, we can use either sines or cosines.

We want to use a sine function here. Right, there's no shift here. But, the graph that's in blue here is actually shifted. Our graph starts at \( t = 0 \) and \( x = 0 \) with a displacement of 3, so we have to write this as \( \sin(kx - \omega t + \phi) \) where \( \phi \) is the phase shift due to the displacement. We want to use \( \sin \) because it starts from zero displacement. If you look at the first hill, the first little crest, it is to the left of where the normal sine graph starts. The wave is shifted to the left from its normal starting point. The sine of our phase constant, \( \sin(\phi) \), equals \(\frac{3}{4}\), and taking inverse sine gives \( \phi = 0.85 \) radians, which we then use in our complete wave function:

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

Now, we're going to repeat this process using a cosine function. A normal cosine graph starts at the amplitude and goes down and up like this. However, our wave starts with a displacement to the right from where a normal cosine starts. We calculate \( \phi_c \) which will be negative in this case because the graph is shifted to the right. Through the same steps, we calculate \( \phi_c = -0.72 \) radians. Our complete wave function using cosine is then:

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

As we've seen, the same wave can be described using either sine or cosine, which will affect the phase constant calculated. This flexibility in wave function notation allows us to adapt to different initial conditions effectively. Let me know if you have any questions.

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. $y\left(x,t\right)=0.15\sin\left(15.7x-50.3t+1.57\right)$

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.