Velocity of longitudinal waves in a solid.

[RazerM]

Homework Statement

Our lecturer gave us a general equation for velocity of waves; (where $c=$ wave velocity)
$$c= \sqrt{\frac{\textrm{springiness}}{\textrm{massiness}}}$$
(Excuse the terms, I'd personally rather have been given the equations here..)

So for transverse waves on a string/wire (where $T=$ Tension and $\mu=$ mass per unit length)
$$c= \sqrt{\frac{T}{\mu}}$$
and for longitudinal waves in a gas (where$B_{ad}=$ the adiabatic bulk modulus and $\rho=$ density)
$$c= \sqrt{\frac{B_{ad}}{\rho}}$$


Where I am stuck is longitudinal waves in a solid, I'm assuming massiness = $\rho$ but am unsure about springiness.

Homework Equations

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The Attempt at a Solution

So for longitudinal waves in a solid; solving for the dimensions of springiness appears to show that springiness is in Newtons but how does Force correlate to a wave through a solid, or am I missing the point completely?

 

[diazona]

hmm, I think you made a mistake in that unit calculation. I get Pascals (N/m^2), a unit of pressure... but I'm not sure what physical quantity that corresponds to, since solids don't really have pressure in the same sense as gasses. This is a particular area of physics in which my knowledge is sadly lacking.

Isn't there a bulk modulus for solids?

 

[kerimek]

The equation provided by your lecturer is a general equation for wave velocity, and it can be applied to different types of waves in various mediums. In the case of longitudinal waves in a solid, the "springiness" refers to the elastic properties of the solid, specifically its Young's modulus (E). This measures the stiffness or rigidity of a solid when subjected to stress or strain. The "massiness" in this case does indeed refer to the density of the solid (ρ).

The equation for the velocity of longitudinal waves in a solid can be written as c = √(E/ρ). This means that the wave velocity is directly proportional to the square root of the elastic modulus and inversely proportional to the square root of the density. This makes sense, as a stiffer and denser solid will have a higher wave velocity compared to a less stiff and less dense solid.

In terms of the units, the elastic modulus (E) is typically measured in Pascals (Pa) and the density (ρ) is measured in kilograms per cubic meter (kg/m^3). When substituted into the equation, the units will cancel out and the resulting unit for wave velocity will be meters per second (m/s).

In summary, the "springiness" in the equation refers to the elastic properties of the solid, specifically the Young's modulus, and the "massiness" refers to the density of the solid. By understanding the physical meaning of these terms and their relationship to wave velocity, we can use the general equation provided by your lecturer to calculate the velocity of longitudinal waves in a solid.