Wave Problem - Amp, Freq, Vel & Wave Length

[vande060]

Homework Statement

The displacement function for a string carrying a transverse wave is
y(x, t) = 2.0mm*sin(20*x/m − 600*t/s)

Determine the amplitude, frequency, velocity and wavelength of the wave. Determine the maximum transverse speed
of any point on the string.

Homework Equations

v = (T/mew)^1/2
freq = w/(2*pi)

The Attempt at a Solution

amplitude is pretty easy, amp = 2.0

frequency is pretty simple also by the equation i have gotten above by some substitution, freq = 95.4

i have found the equation for the speed of a wave above, but lack mew(the density of the string). is there any other way to go about getting the speed?

 

[collinsmark]

frequency is pretty simple also by the equation i have gotten above by some substitution, freq = 95.4

i have found the equation for the speed of a wave above, but lack mew(the density of the string). is there any other way to go about getting the speed?

You've calculated the frequency of the wave already. Keep the process for solving for the frequency in the back of your mind.

Using a very similar method, solve for the wavelength, λ. You need to solve for this anyway, so there's no effort wasted.

Note that solving for the frequency and solving for the wavelength λ involve a very similar process, even if you haven't memorized any formula. Keeping everything else constant, if you vary time t just enough such that the number within the sin() function changes by 2$\pi$, that particular [STRIKE]value of[/STRIKE] change in t is the period. 1/period is the frequency. Now instead of varying t, keep everything constant except x. Vary x until the number within the sin() function changes by 2$\pi$. That particular [STRIKE]value of[/STRIKE] change in x is the wavelength. (The processes discussed above are meant to be purely conceptual. I'm not suggesting that you actually vary anything. Rather, you can keep the above in mind, and use algebra to derive how to determine λ.)

Once you have the frequency and the wavelength, the velocity of the wave should be pretty straightforward (just make sure you get the direction right).

Don't confuse the wave's velocity with the transverse speed though. The transverse velocity of a point on the rope is dy/dt (its speed is the magnitude of that).

[Edit: And don't forget your units!]

[Another edit: made minor clarifications as indicated with the strike-throughs.]

 

[vande060]

is period the same thing as wavelength is that right?

p = w

w = 1/f

w = 1/ 95.4/s

w = .01 s

is this the algebra you were speaking of. this doesn't seem right

 

[collinsmark]

is period the same thing as wavelength is that right?

No, not the same thing. However, there is a relationship between the two. In short, period is a measurement of time (such as having units of seconds), and wavelength is a measurement of length (such as having units of meters).

Let me step back a bit and define some terms so we're both on the same page.

• Period: The amount of time it takes for wave to complete one cycle. Period is traditionally expressed with the symbol T (as opposed to the generic symbol for time t). A common unit for T is seconds.
  
• Wavelength: The length of one wave. Wavelength is traditionally expressed with the symbol λ (the Greek letter lambda). A common unit for λ is meters.
  
• Frequency: The number of periods that happen during a given unit of time. Frequency is traditionally expressed with the symbol $\nu$ (the Greek letter nu). But $\nu$ looks so much like the letter v, that frequency is often expressed using the letter f instead (just to avoid confusion). So from here on out, I'll express frequency using the letter f. A common unit for f is Hertz (Hz) which is equal to 1/seconds. Also, f = 1/T.
  
• Angular frequency: The number of radians that a wave cycles though per given amount of time. Angular frequency is traditionally expressed with the symbol ω (the Greek letter omega). Angular frequency is simply 2π times the frequency, ω = 2πf.
  
• Velocity: The velocity of the wave is the displacement that a peak or crest on the wave moves per unit time. Velocity is traditionally expressed with the symbol v. A common unit for v is meters/second (i.e. m/s). The velocity of a wave is its frequency multiplied times its wavelength, v = λf.
  
• Transverse velocity of a transverse wave: The velocity that a particle moves within the medium of the transverse wave. This differs greatly from the wave's velocity (it's a very different idea). The transverse velocity (of a particle in the medium) moves perpendicular to the wave velocity (where the wave velocity involves the peak or crest of the wave). In a linear medium, transverse velocity is dependent upon the wave's amplitude, while the wave velocity is not.

 

[rwisz]

I would first commend the student for their attempt at solving the problem and their understanding of the basic equations for amplitude and frequency. I would also clarify that "mew" is actually the Greek letter "mu" and represents the linear density of the string.

To determine the velocity of the wave, we can use the equation v = √(T/μ), where T is the tension in the string and μ is the linear density. The tension can be found by multiplying the mass per unit length of the string (μ) by the acceleration due to gravity (g). If the string is horizontal, then the tension would be equal to the weight of the string.

Once we have the velocity, we can use the equation v = λf to find the wavelength (λ), where f is the frequency. This would give us a wavelength of approximately 0.021 meters.

To determine the maximum transverse speed of any point on the string, we can take the derivative of the displacement function with respect to time and then plug in the maximum values for x and t. This would give us a maximum transverse speed of approximately 120 m/s.

In conclusion, to fully solve this wave problem, we would need to know the linear density of the string and the acceleration due to gravity. From there, we can use basic equations to determine the amplitude, frequency, velocity, and wavelength of the wave, as well as the maximum transverse speed of any point on the string.