# Expressing horizontal velocity as a function of time for a wave

• JoeyBob
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).
JoeyBob
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?

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).

## 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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