Energy in Waves: Intensity Learning Objectives By the end of this section, you will be able to: Calculate the intensity and the power of rays and waves.
Example 1. Calculating intensity and power: How much energy is in a ray of sunlight? Calculate the amount of energy that falls on a solar collector having an area of 0. What intensity would such sunlight have if concentrated by a magnifying glass onto an area times smaller than its own? Calculate to find E and convert units: 5. Discussion for Part 1 The energy falling on the solar collector in 4 h in part is enough to be useful—for example, for heating a significant amount of water.
Strategy for Part 2 Taking a ratio of new intensity to old intensity and using primes for the new quantities, we will find that it depends on the ratio of the areas. Discussion for Part 2 Decreasing the area increases the intensity considerably. Example 2. Determine the combined intensity of two waves: Perfect constructive interference If two identical waves, each having an intensity of 1. Strategy We know from Superposition and Interference that when two identical waves, which have equal amplitudes X , interfere perfectly constructively, the resulting wave has an amplitude of 2 X.
Solution Recall that intensity is proportional to amplitude squared. Discussion Figure 2. Conceptual Questions Two identical waves undergo pure constructive interference. Is the resultant intensity twice that of the individual waves? Explain your answer. Circular water waves decrease in amplitude as they move away from where a rock is dropped.
Explain why. Ultrasound of intensity 1. What is its power output? The low-frequency speaker of a stereo set has a surface area of 0. What is the intensity at the speaker?
If the speaker projects sound uniformly in all directions, at what distance from the speaker is the intensity 0. To increase intensity of a wave by a factor of 50, by what factor should the amplitude be increased?
Engineering Application. A device called an insolation meter is used to measure the intensity of sunlight has an area of cm2 and registers 6. Astronomy Application. How long does it take for 1. Suppose you have a device that extracts energy from ocean breakers in direct proportion to their intensity.
To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. This kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave:. There is also potential energy associated with the wave.
Much like the mass oscillating on a spring, there is a conservative restoring force that, when the mass element is displaced from the equilibrium position, drives the mass element back to the equilibrium position. Integrating over the wavelength, we can compute the potential energy over a wavelength:.
The potential energy associated with a wavelength of the wave is equal to the kinetic energy associated with a wavelength. The total energy associated with a wavelength is the sum of the potential energy and the kinetic energy:. The time-averaged power of a sinusoidal mechanical wave, which is the average rate of energy transfer associated with a wave as it passes a point, can be found by taking the total energy associated with the wave divided by the time it takes to transfer the energy.
If the velocity of the sinusoidal wave is constant, the time for one wavelength to pass by a point is equal to the period of the wave, which is also constant. For a sinusoidal mechanical wave, the time-averaged power is therefore the energy associated with a wavelength divided by the period of the wave. The wavelength of the wave divided by the period is equal to the velocity of the wave,. Note that this equation for the time-averaged power of a sinusoidal mechanical wave shows that the power is proportional to the square of the amplitude of the wave and to the square of the angular frequency of the wave.
Consider a two-meter-long string with a mass of The tension in the string is When the string vibrator is turned on, it oscillates with a frequency of 60 Hz and produces a sinusoidal wave on the string with an amplitude of 4. What is the time-averaged power supplied to the wave by the string vibrator? The power supplied to the wave should equal the time-averaged power of the wave on the string.
The speed of the wave on the string can be derived from the linear mass density and the tension. The string oscillates with the same frequency as the string vibrator, from which we can find the angular frequency. The amplitude is given, so we need to calculate the linear mass density of the string, the angular frequency of the wave on the string, and the speed of the wave on the string.
This is true for most mechanical waves. If either the angular frequency or the amplitude of the wave were doubled, the power would increase by a factor of four. The time-averaged power of the wave on a string is also proportional to the speed of the sinusoidal wave on the string.
If the speed were doubled, by increasing the tension by a factor of four, the power would also be doubled. Is the time-averaged power of a sinusoidal wave on a string proportional to the linear density of the string? The equations for the energy of the wave and the time-averaged power were derived for a sinusoidal wave on a string. In general, the energy of a mechanical wave and the power are proportional to the amplitude squared and to the angular frequency squared and therefore the frequency squared.
Another important characteristic of waves is the intensity of the waves. Waves can also be concentrated or spread out. Waves from an earthquake, for example, spread out over a larger area as they move away from a source, so they do less damage the farther they get from the source.
Changing the area the waves cover has important effects. All these pertinent factors are included in the definition of intensity I as power per unit area:. The definition of intensity is valid for any energy in transit, including that carried by waves.
Many waves are spherical waves that move out from a source as a sphere. For example, a sound speaker mounted on a post above the ground may produce sound waves that move away from the source as a spherical wave. Sound waves are discussed in more detail in the next chapter, but in general, the farther you are from the speaker, the less intense the sound you hear. If there are no dissipative forces, the energy will remain constant as the spherical wave moves away from the source, but the intensity will decrease as the surface area increases.
In the case of the two-dimensional circular wave, the wave moves out, increasing the circumference of the wave as the radius of the circle increases. If you toss a pebble in a pond, the surface ripple moves out as a circular wave. As the ripple moves away from the source, the amplitude decreases. Consider a string with under tension with a constant linear mass density.
A high energy wave is characterized by a high amplitude; a low energy wave is characterized by a low amplitude. As discussed earlier in Lesson 2 , the amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position.
The logic underlying the energy-amplitude relationship is as follows: If a slinky is stretched out in a horizontal direction and a transverse pulse is introduced into the slinky, the first coil is given an initial amount of displacement. The displacement is due to the force applied by the person upon the coil to displace it a given amount from rest. The more work that is done upon the first coil, the more displacement that is given to it. The more displacement that is given to the first coil, the more amplitude that it will have.
So in the end, the amplitude of a transverse pulse is related to the energy which that pulse transports through the medium. Putting a lot of energy into a transverse pulse will not effect the wavelength, the frequency or the speed of the pulse. The energy imparted to a pulse will only affect the amplitude of that pulse.
Consider two identical slinkies into which a pulse is introduced. If the same amount of energy is introduced into each slinky, then each pulse will have the same amplitude. But what if the slinkies are different? What if one is made of zinc and the other is made of copper? Will the amplitudes now be the same or different? If a pulse is introduced into two different slinkies by imparting the same amount of energy, then the amplitudes of the pulses will not necessarily be the same.
In a situation such as this, the actual amplitude assumed by the pulse is dependent upon two types of factors: an inertial factor and an elastic factor. Two different materials have different mass densities. The imparting of energy to the first coil of a slinky is done by the application of a force to this coil.
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