Expressing horizontal velocity as a function of time for a wave

In summary, to find the horizontal velocity as a function of time for the given wave function, you take its partial t derivative and insert x=4, which gives a function of -2.4sin(1-12t). This is because the horizontal velocity is constant and not dependent on x or t. The partial t derivative represents the change in the vertical velocity at a given (x,t) and the partial x derivative represents the slope at a given (x,t).
  • #1
JoeyBob
256
29
Homework Statement
y(x, t) = 0.2 sin(0.25x-12t). Find horizontal velocity as a function of time at x=4
Relevant Equations
none
This is more of a conceptual question. To find the horizontal velocity as a function of time for the above wave function, you take its partial t derivative and insert x=4. In other words the function would be -2.4sin(1-12t).

Im wondering why you take the partial t derivative and not to partial x derivative? I understand how partial derivatives work, but I still don't think I am entirely sure what they represent. Like, why can't you take the partial x derivative? Wouldnt the change in x be about velocity? What does a partial t derivative actually mean / represent in this context? Horizontal velocity, but why/how?
 
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  • #2
JoeyBob said:
Homework Statement:: y(x, t) = 0.2 sin(0.25x-12t). Find horizontal velocity as a function of time at x=4
Relevant Equations:: none

This is more of a conceptual question. To find the horizontal velocity as a function of time for the above wave function, you take its partial t derivative and insert x=4. In other words the function would be -2.4sin(1-12t).

Im wondering why you take the partial t derivative and not to partial x derivative? I understand how partial derivatives work, but I still don't think I am entirely sure what they represent. Like, why can't you take the partial x derivative? Wouldnt the change in x be about velocity? What does a partial t derivative actually mean / represent in this context? Horizontal velocity, but why/how?
Maybe some more explanation is needed regarding "horizontal" and x, y.
I would have taken a wave written as y(x,t) to mean x is horizontal and y is vertical. In that case, taking partial wrt t gives vertical velocity at a given (x,t).

Taking partial wrt x gives the slope at a given (x,t).

The horizontal velocity would be a constant, not dependent on x or t. If you increase t by Δt then that decreases the argument of the sin by 12Δt, so compensate and get the same value of the argument you need to increase x by 12Δt/0.25=48Δt. So velocity is 48 (of whatever units were used for the 0.25 and 12).
 

Related to Expressing horizontal velocity as a function of time for a wave

1. What is horizontal velocity in relation to a wave?

Horizontal velocity is the speed at which a wave moves horizontally, or side to side. It is a measure of how quickly the wave is propagating through a medium.

2. How is horizontal velocity expressed as a function of time for a wave?

Horizontal velocity can be expressed as a function of time using the equation v = λ/T, where v is the horizontal velocity, λ is the wavelength, and T is the period of the wave.

3. What factors can affect the horizontal velocity of a wave?

The horizontal velocity of a wave can be affected by the properties of the medium it is traveling through, such as density and elasticity, as well as external factors like temperature and pressure.

4. Can horizontal velocity change as a wave travels?

Yes, horizontal velocity can change as a wave travels through different mediums. It can also change if the properties of the medium it is traveling through change, such as a change in temperature or pressure.

5. How is horizontal velocity related to other properties of a wave?

Horizontal velocity is related to other properties of a wave, such as frequency, wavelength, and amplitude. These properties are all interconnected and can affect each other as a wave travels through a medium.

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