Wave speed equation problem

In summary: Written in the form of units L/T^2.Now you have your units for the equation. To check that your units are right, you can compare them to the units for velocity, which are L/T. Since the units for a) do not match the units for velocity, this equation is incorrect. In summary, the speed of waving in shallow water depends on the acceleration of gravity and the water depth. The correct formula for wave speed is v=√gh, which is derived through dimensional analysis. The incorrect formula, v=1/2gh^2, does not match the units for velocity.
  • #1
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Homework Statement


The speed of waving in shallow water depend only on the acceleration of gravity g, a quantity with dimensions L/T^2, and on the water depth h.

Which of the following formulas for the wave speed v could be correct?

a) v=1/2gh^2 b)v=√gh


Homework Equations


L = [tex]\frac{n\lambda_{n}}{2}[/tex]

v[tex]^{2}[/tex] = [tex]\frac{T}{\frac{m}{l}}[/tex]

v = f[tex]\lambda[/tex]


The Attempt at a Solution



I think I have to use one of above equations, but I don't know where to start.
 
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  • #2
I think you are over complicating things. It's a simple dimensional analysis problem. Your a) equation comes out in units of m3/s2 while your b) equation comes out in m/s. Velocity has units of m/s, therefore equation b) is correct.

Actually deriving this equation for a water wave involves the inverse hyperbolic tangent function.
 
  • #3
could you explain how the equation a) comes out to be m3/s2 because I don't understand the process.
 
  • #4
So for a) you have 0.5gh2

Water Depth is measured in SI units of meters. When you square meters it becomes meters squared or more generally length squared.

"g" or the acceleration due to gravity has units of meters per second squared. Or more generally Length per unit time squared (written L/T^2)

0.5 does not have units so you can leave it out in a dimensional analysis problem.

Now multiply the units from h and g together.

Square meters times meters gives you cubic meters just like x2 times x gives you x3.

Seconds squared stays seconds squared.

More generally it comes out length cubed over time squared.
 

What is the wave speed equation?

The wave speed equation is a mathematical formula that relates the speed of a wave to its wavelength and frequency. It is commonly written as v = λf, where v is the wave speed, λ is the wavelength, and f is the frequency.

How do you calculate wave speed using the wave speed equation?

To calculate wave speed using the wave speed equation, you need to know the wavelength and frequency of the wave. Simply plug these values into the equation v = λf and solve for v. The resulting value will be the speed of the wave in the given medium.

What is the unit of measurement for wave speed?

The unit of measurement for wave speed depends on the units used for wavelength and frequency. If wavelength is measured in meters (m) and frequency is measured in hertz (Hz), then the unit of measurement for wave speed will be meters per second (m/s).

What factors affect wave speed?

The speed of a wave can be affected by a variety of factors, including the properties of the medium through which it is traveling, the amplitude of the wave, and the temperature of the medium. In general, waves tend to travel faster in denser mediums and slower in less dense mediums.

Can the wave speed equation be used for all types of waves?

Yes, the wave speed equation can be used for all types of waves, including electromagnetic waves, sound waves, and water waves. However, for different types of waves, the values for wavelength and frequency may be measured differently. For example, sound waves have a frequency measured in hertz, while electromagnetic waves have a frequency measured in cycles per second (cps) or oscillations per second (ops).

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