Question

Derive the expression for the energy of a wave \(E=\frac{1}{2} m \omega^{2} A^{2},\) where \(m\) is the mass of the substance that is oscillating, \(\omega\) is the angular velocity, and \(A(r)\) is the amplitude. Start by expressing the kinetic energy and potential energy of the wave, and then add these to get the energy of the wave.

Short answer

Answer: The expression for the energy of a wave is given by \(E=\frac{1}{2}m \omega^{2}A^{2}\), where \(E\) represents energy, \(m\) represents mass, \(\omega\) represents angular velocity, and \(A\) represents amplitude.

Step-by-step solution

Expressing the kinetic energy of a wave

According to classical mechanics, the kinetic energy (K.E.) of an oscillating object is given by the formula: $$K.E.= \frac{1}{2} m v^2$$ Where \(m\) represents the mass of the substance and \(v\) is its velocity. For a wave, the velocity is given by the product of its angular velocity, \(\omega\), and amplitude, \(A\): $$v = \omega A$$ Now, substitute the expression for velocity into the kinetic energy formula: $$K.E.= \frac{1}{2} m (\omega A)^2$$

Expressing the potential energy of a wave

The potential energy (P.E.) of a wave can be given by the formula for the energy stored in a spring: $$P.E.=\frac{1}{2} k x^2$$ Here, \(k\) is the spring constant, and \(x\) is the displacement from the equilibrium position, which is equal to the amplitude \(A(r)\). Since \(k = m\omega^2\), we can rewrite the potential energy formula as: $$P.E.= \frac{1}{2} m\omega^2 A^2$$

Adding kinetic and potential energy to get the total energy

Now, we will add the kinetic energy and potential energy to get the total energy of the wave: $$E = K.E. + P.E.$$ Using the expressions from Steps 1 and 2, we get: $$E = \frac{1}{2} m (\omega A)^2 + \frac{1}{2} m\omega^2 A^2$$ Simplify the expression: $$E=\frac{1}{2}m \omega^{2}A^{2}$$ This is the expression for the energy of a wave, where \(E\) represents energy, \(m\) represents mass, \(\omega\) represents angular velocity, and \(A\) represents amplitude.

Key concepts

Kinetic Energy

Understanding the kinetic energy of a wave is essential when analyzing wave motion. Kinetic energy is the energy an object has due to its motion. It can be calculated with the equation:
\[ K.E. = \frac{1}{2} m v^2 \]
where \(m\) is the mass of the object and \(v\) is its velocity. In the context of a wave, this velocity is not constant but instead changes throughout the wave's oscillation. Specifically, for a point on the wave, its maximum velocity is the product of angular velocity \(\omega\) and amplitude \(A\), represented by:
\[ v = \omega A \]
When a wave oscillates, every point on it experiences kinetic energy changes due to velocity changes. Substituting the velocity of a wave into the kinetic energy formula shows the direct relationship between kinetic energy, angular velocity, and amplitude.

Potential Energy

The potential energy in wave motion plays a role similar to that in a compressed or stretched spring. It is the energy stored due to the position of the particles within the wave. The potential energy in a wave is given by:
\[ P.E. = \frac{1}{2} k x^2 \]
In this equation, \(k\) is analogous to the spring constant and represents the restoring force strength, and \(x\) is the displacement, equivalent to the wave's amplitude \(A\). However, when dealing with waves, we consider the mass \(m\) and angular velocity \(\omega\), where \(k = m\omega^2\). This allows us to express the potential energy as a function of angular velocity and amplitude, showing how a wave's stored energy varies with these properties.

Angular Velocity

Angular velocity \(\omega\) is a measure of how quickly an object rotates or oscillates around a fixed point or axis. It is a key component in determining the energy within a wave as it directly affects both kinetic and potential energies. Specifically:
\[ v = \omega A \]
illustrates that the instantaneous velocity of a point on a wave is a product of angular velocity and the amplitude of the wave. As angular velocity increases, so does the velocity of each point on the wave, resulting in higher kinetic energy. Moreover, angular velocity also factors into potential energy since the spring constant-like term \(k\) can be expressed as \(m\omega^2\), entwining angular velocity with the energy stored in the wave's compression or expansion.

Amplitude

The amplitude \(A\) of a wave is a critical parameter that represents the maximum displacement of a point on the wave from its rest position. It plays a dual role in the energy equation of a wave by affecting both kinetic and potential energy forms. For kinetic energy, amplitude, when combined with angular velocity, defines the maximum speed of the oscillating particles:
\[ v = \omega A \]
while for potential energy, it acts as the displacement in the classical spring equation:
\[ P.E. = \frac{1}{2} k x^2 \]
This displacement, directly linked to amplitude, shows that larger amplitudes result in greater energy storage in the system during a wave's oscillation. The overall energy of a wave is thus intrinsically linked to how greatly the particles within the wave are displaced during their motion.

Oscillation

Oscillation refers to the repetitive movement of particles in a wave around an equilibrium position. This movement is the source of both kinetic and potential energy in a wave as particles oscillate, they continually convert energy from kinetic to potential form and vice versa. The frequency of oscillation determines the number of times a wave cycles through these energy conversions in a given period, while the amplitude and angular velocity determine the extent of kinetic and potential energy in each cycle. Through oscillation, we see the dynamic nature of wave energy, with the particles' back and forth motion reflecting the interplay between kinetic and potential forms.