# Wave equation - v speed or velocity?

• spaghetti3451
In summary, the classical wave equation involves the quantity "v" which is typically referred to as the wave velocity, but can also be interpreted as the wave speed. The direction of v is not specified in the equation itself, but rather in the boundary conditions. V^2 is a scalar quantity and can be expressed in terms of either speed or velocity. The dispersion relation in the wave equation is given by v = |k|, where v is a scalar quantity known as the phase velocity. In more complex situations, such as in anisotropic media, the wave equation can become more complicated.

#### spaghetti3451

Consider the classical wave equation.

Does the v in the equation stand for the wave speed or the wave velocity? The image says v is the wave velocity but I am not sure about that as other sources like Wikipedia say that v is the wave speed. So, what is v really - speed or velocity?

I think it's wave velocity since v is squared, so that allows for the possibility for v being positive or negative. And the direction of travel is the direction of the wavevector.

What do you think?

I think that when v is squared, there is no distinction. You can use vector notation where the "wave vector" k is a vector in the direction of propagation, but the wave equation results for any direction for k, the actual direction appears in the boundary conditions to the equation, not in the equation itself. So you can interpret v as having a specified direction that the equation doesn't care about, or you can imagine that v has no specified direction until it emerges from the boundary conditions-- the equation is the same either way.

Here, it's of course the speed, since it enters only as $v^2$, which is a scalar (under rotations).

Ken G said:
I think that when v is squared, there is no distinction. You can use vector notation where the "wave vector" k is a vector in the direction of propagation, but the wave equation results for any direction for k, the actual direction appears in the boundary conditions to the equation, not in the equation itself. So you can interpret v as having a specified direction that the equation doesn't care about, or you can imagine that v has no specified direction until it emerges from the boundary conditions-- the equation is the same either way.

I see. I did not solve wave equations before, so I had no idea that boundary conditions are needed to specify the complete problem. So, thanks for the insight.

vanhees71 said:
Here, it's of course the speed, since it enters only as $v^2$, which is a scalar (under rotations).

But that contradicts Ken G, doesn't it?

I have no idea how $v^2$ is a scalar under rotations. Do you mean that if you imagine a coordinate system with three degrees of freedom for velocity, then if you rotate $v^2$ in the coordinate system, $v^2$ remains the same?

$v^2$ is just a number. It doesn't matter if you got it by taking $|\vec{v}|^2$ or by $\vec{v}\cdot\vec{v}$. The result is the same.

Matterwave said:
$v^2$ is just a number. It doesn't matter if you got it by taking $|\vec{v}|^2$ or by $\vec{v}\cdot\vec{v}$. The result is the same.

I see! So, you agree with Ken G that we have the freedom to decide if we should take the quantity as either speed or velocity?

So, vanhees71 is wrong, I suppose?

Of course $v^2=\vec{v}^2$ wherever you get a vector quantity from. The wave equation reads
$\frac{1}{v^2} \frac{\partial^2 \phi}{\partial t^2}-\Delta \phi=0.$
Any solution can be expressed via plain waves
$$\phi(t,\vec{x})=\int_{\mathbb{R}^3} \frac{\mathrm{d}^3 \vec{k}}{(2 \pi)^3} \exp[-\mathrm{i} \omega(\vec{k}) t] \left [A(\vec{k}) \exp[+\mathrm{i} \vec{k} \cdot \vec{x}] + B(\vec{k}) \exp[-\mathrm{i} \vec{k} \cdot \vec{x}] \right ].$$
$$\omega(\vec{k})=v |\vec{k}|.$$
As you see again from the very beginning of the derivation, $v$ is a scalar quantity! It's usually called the phase velocity of the wave although it's a scalar quantity.

Of course, this equation only applies in homogeneous and istropic media. In more complicated cases you have more complicated wave equations (like for light propagation in an anisotropic medium like crystals).

## 1. What is the wave equation?

The wave equation is a mathematical formula that describes the behavior of a wave as it propagates through a medium. It is used to calculate the displacement, velocity, and acceleration of a wave at any given point in space and time.

## 2. How is velocity or speed related to the wave equation?

The wave equation includes the variable of velocity, which represents the speed at which the wave travels through a medium. This means that the wave equation can be used to determine the velocity of a wave based on its wavelength and frequency.

## 3. What factors affect the speed of a wave?

The speed of a wave is affected by the properties of the medium through which it is traveling. For example, the density and elasticity of the medium can impact the speed of a wave. In addition, the wavelength and frequency of the wave can also affect its velocity.

## 4. Can the wave equation be applied to all types of waves?

Yes, the wave equation can be applied to all types of waves, including electromagnetic waves, sound waves, and water waves. However, the specific variables and constants used in the equation may vary depending on the type of wave being studied.

## 5. How is the wave equation used in real-world applications?

The wave equation has many practical applications, such as in the fields of acoustics, optics, and seismology. It is used to design and analyze various technologies, such as ultrasound machines, fiber optics, and earthquake-resistant buildings. It is also used in scientific research to study the behavior of waves in different mediums.