Wave Formula

A wave simply refers to a transfer of energy through a medium. The parachutes tend to make terrific ripples that act quite similar to waves. The ocean waves receive power from the wind. The water takes on this energy and transfers it through a wave. This wave can certainly carry on for long distances. One can think of waves as swells and depressions. Furthermore, the waves are not restricted to water only. Most noteworthy, the powering of the waves can take place in plenty of ways like the shifting of the plates of the Earth or sound production. Learn wave formula here.

What is Wave

A wave refers to a disturbance of a field in which the oscillation of a physical attribute takes place repeatedly at each point. Furthermore, when in a wave, a physical attribute seems appears to move through space. Moreover, the waves which are common in the field of physics are electromagnetic and mechanical. Some other types of waves include the gravitational waves, physical waves, and plane waves.

Wave Formula

The production waves take place when a vibrating source disturbs the first particle of a periodical nature. This certainly creates a wave pattern that begins to travel from particle to particle along with a medium. Furthermore, the frequency at the vibration of a particle occurs and is equal to the frequency of sound vibration.

Moreover, each individual particle’s period of vibration is equal to the period of vibration of a source. Most noteworthy, one complete wave cycle refers to the complete back and forth movement.

In a time of one period, the movement of the wave is one wavelength. Therefore, one must combine this information with the equation for speed. Consequently, one can say that the speed of a wave is certainly the wavelength/period.

Speed = Wavelength/Period

Also, the period refers to the reciprocal of the frequency. The substitution of the expression 1/f can take place into the above equation for period. Therefore, the rearrangement of the equation gives a new equation of the form:

Speed = Wavelength × Frequency

The above equation or formula is the waves equation. It gives the mathematical relationship between speed of a wave and its wavelength and frequency. Therefore, the equation or formula can rewritten as

v = f × λ
where, v = speed of
f = frequency
λ = wavelength

Wave Formula Derivation

The derivation of the wave equation certainly varies depending on context. There is a particular simple physical setting for the derivation. Moreover, this setting is that of small oscillations on a piece of string in accordance with the Hooke’s law.

One must consider the forces acting on a small element of particular mass dm contained in a small interval dx. In case of a small displacement, the horizontal force is approximately zero. The vertical force happens to be

\(\Sigma F_{y}\) = \({T}’ sin\Theta_{2}\) – T \(sin\Theta_{1}\) = (dm)a = \(\mu dx \frac{\delta ^2 y}{\delta t^2}\)

Here, µ refers to the mass density, \( \mu =  \frac{\delta m}{\delta x} \)

On the other hand,  the horizontal force is approximately zero when displacements are small, \( T cos \theta^{1} \approx T{}’cos \theta_{2} \approx T \). So,

\(-\frac{\mu dx\frac{\delta^{2}y}{\delta t^{2}}}{T}\) \( \approx \) \(\frac{T{}’sin \theta_{2 + T sin \theta_{1}}}{T}\) = \(\frac{{T}’sin\theta_{1}}{T}\) + \(\frac{T sin\theta_{1}}{T}\) \(\approx \frac{{T}’ sin\theta_{2}}{T{}’ cos}\) + \(\frac{T sin\theta_{1}}{T cos\theta_{1}}\) = \(tan\theta_{1} + tan\theta_{2}\)

But, \(tan \theta_{1}\) + \(tan \theta_{2}\) = \(-\Delta \frac{\delta y}{\delta x}\), the difference happens to be between x and x + dx. This is because, the tangent is certainly equal to the slope in a geometrical manner. So, if one divides over dx, one finds

\(-\frac{\mu \frac{\delta^{2}y}{\delta t^{2}}}{T}\) = \(\frac{tan \theta_{1} + tan \theta_{2}}{dx}\) = \(-\frac{\Delta \frac{\delta y}{\delta x}}{dx}\)

The rightmost term above refers to the definition of the derivative with respect to x. This is because, the difference is over an interval dx, and therefore one has

\(\frac{\mu }{T} \frac{\delta ^{2}y}{\delta t^{2}}\) = \(\frac{\delta ^{2}y}{\delta x^{2}}\)

which happens to be exactly the wave equation in one dimension for velocity v = \(\sqrt{\frac{T}{\mu }}\)

Solved Examples on Wave Formula 

Q1 A wave has a frequency of 50 Hz. It also has a wavelength of 10 m. Find out the speed of the wave?
A1 We have:

f = 50 Hz
λ =10 m
v = ?
v = λ× f = (10 m)× (50 Hz) = 500 m/s

Therefore, speed of the wave is 500 m/s.

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