Waves - Physics Textbook Chapter with Doppler Effect and Interference - PDF

Okay, here is a structured markdown version of the provided text, with LaTeX formatting for formulas and descriptions for images: # WAVES A wave is a disturbance that moves through a medium or through space from the place it was created. For water waves, the disturbance is on the surface of the wa...

Okay, here is a structured markdown version of the provided text, with LaTeX formatting for formulas and descriptions for images: # WAVES A wave is a disturbance that moves through a medium or through space from the place it was created. For water waves, the disturbance is on the surface of the water, perhaps created by a rock thrown into a pond or by a swimmer splashing the surface repeatedly. For sound waves, the disturbance is a change in air pressure, perhaps created by the oscillating cone inside a speaker. For earthquakes, there are several types of disturbances, including disturbance of Earth's surface and pressure disturbances under the surface. The world is full of waves: sound waves, waves on a string, seismic waves, and electromagnetic waves, such as visible light, radio waves, television signals, and x-rays. All these waves have their source, a vibrating object. The importance of radio and television signals and other forms of electromagnetic waves cannot be ignored. Communication using these phenomena is the backbone of modern civilization. Many waves, such as sound and light, transfer energy from one location to another. In this chapter, first we focus on the fact that waves can combine with one another, reflect off the surfaces and absorbed by materials they enter. All these effects vary with wavelength. Mechanical waves transport energy through a medium. Electromagnetic radiation is differentiated by wavelength or frequency, and includes radio waves, microwaves, infrared, visible light, ultraviolet radiation, x-rays, and gamma rays. These wavelengths vary from radio waves (the longest) to gamma rays (the shortest). In empty space all electromagnetic waves move at the same speed, the "speed of light". In this chapter we will also study about different phenomenon of waves such as stationary waves and Doppler effect and its applications. ## 9.1 INTENSITY OF WAVES All waves carry energy and sometimes their energy can be directly observed. For example, earthquakes can shake whole cities, performing the work of thousands of wrecking balls. Loud sounds beat nerve cells in the inner ear, causing permanent hearing loss. Ultrasound is used for deep-heat treatment of muscle strains. A laser beam can burn away a malignancy. Water waves abolish beaches. The amount of energy in a wave is related to the square of its amplitude, as shown in the following relation: $Intensity \propto (amplitude)^2$ So an earthquake having large amplitude produces large ground displacements. The energy of a wave depends on time as well. For example, the longer deep-heat ultrasound is applied, the more energy it transfers. Waves can also be concentrated or spread out. For example, sunlight can be focused to burn wood. Earthquakes spread out, so they do less damage as farther they get from the source. In both cases, 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: $I = \frac{P}{A}$ (9.1 a) where P is the power carried by the wave through area A. As, power is energy per unit time (P = E/t), equation (9.1 a) can be written as: $I = \frac{E}{A \times t}$ (9.1 b) The definition of intensity is valid for any energy in transit, including that carried by progressive waves. The SI unit for intensity is watts per square meter (W/m²). **EXAMPLE 9.1:** 1. The average intensity of sunlight on Earth's surface is about 500 W/m². Calculate the amount of energy that falls on a solar collector having an area of 0.50 m² in 4.0 h. * **Given:** * Intensity of sunlight on the Earth = I = 500 W/m² * Area of solar collector = A = 0.5 m² * Time = t = 4.0 h * **To Find:** * Energy fall on solar collector = E = ? * **Solution:** Using equation: $I = \frac{E}{A \times t}$ $E = I \times A \times t$ Substitute the values into the equation, we get: $E = (500 W/m^2)(0.50 m^2)(4.0 \times 3600 s)$ $E = 3.6 \times 10^6 J$ Hence, the energy falling on the solar collector in 4 h is enough for heating a significant amount of water. 2. If amplitude of the wave is doubled then how much energy is increased? * **Solution:** Energy depends on the intensity of the wave. Intensity is proportional to the square of the amplitude. i.e., $Intensity \propto (amplitude)^2$ If amplitude is increased to double then energy will be increased to 4 times because 4 is the square of 2. *** **Assignment 9.1** To increase intensity of a wave by a factor of 50, by what factor should the amplitude is increased? ## 9.2 DOPPLER'S EFFECT Possibly you have noticed how the sound of a vehicle's horn changes as the vehicle moves past you. The frequency of the sound you hear acts higher as the vehicle approaches you and lower as it moves away from you. This phenomenon is one example of the Doppler's Effect, named for Austrian physicist Christian Doppler (1803-1853), who discovered it in 1842. The apparent change in the frequency of sound due to the relative motion between the listener and source of sound is called Doppler's effect. Although the Doppler's effect is most often associated with sound, it's common to all waves, including water and light. In deriving the Doppler's effect, we assume the air is stationary and all speed measurements are made relative to this stationary medium. Motion of a source of sound toward an observer increases the rate at which he or she receives the vibrations. The velocity of each vibration is the speed of sound whether the source is moving or not. Each vibration from an approaching source has a shorter distance to travel. The wavelength is shortened when the source is moving toward the observer and is lengthened when the source is moving away from the observer. The vibrations are therefore received at a higher frequency than they are sent. Similarly, sound waves from a receding source are received at a lower frequency than that at which they are sent. Consider a source of sound S emitting sound wave of velocity *v*, frequency *f* and wavelength $\lambda$. When source S and listener L are at rest then listener will receive f number of waves per second, according to relation: $v = f \lambda$ $f = \frac{v}{\lambda}$........ (i) Following are the possible situations regarding source of sound and the listener. **(1) The Source is at Rest and Listener is Moving** **(a) Listener Moves Toward Stationary Source:** Let the listener L is moving with speed 𝑣L towards the stationary sounding source S as shown in figure 9.1(a). The speed of sound relative to listener is increases to (ν + νL) and wavelength remains unchanged. *Description for the image of Figure 9.1 (a): Shows a figure of Listener moving towards a stationary audible wave source.* The apparent frequency f' is: $f' = \frac{(ν+ νL)}{\lambda}$ Putting $\lambda = \frac{v}{f}$, we get: $f' = \frac{v + vL}{v} f (9.2)$ As $\frac{v + vL}{v} > 1$ So, $f' > f$ Hence, the frequency of sound increases, as a result its pitch also increases. **(b) Listener Moves Away from Stationary Source:** Let the listener L is moving away from the stationary sounding source S with speed vL, then speed of sound relative to listener is decreases to (v – vL) and wavelength remains unchanged. The apparent frequency f 'is: $f' = \frac{V - VL}{\lambda}$ Putting $\lambda = \frac{v}{f}$, we get $ f' = \frac{V-VL}{V} f (9.3) $ Figure 9.1 (b) *Shows a figure of Listener moving away from a stationary audible wave source.* As $\frac{V-VL}{V} < 1$ Hence, the frequency of sound decreases results itch decreases \(2) The Source of moving and Listererer is at the rest (a) when Source Moves Toward Stationary Listener if source moving speed VS toward the Stationary Listener A shown as figure, then each new wave emitted from a position from left to right is on the source F' = v/ = (y-vs)/f ..... (I) $ Vs The apperant the f = ...... 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