# Lesson 19: Mechanical Waves!!

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1 Lesson 19: Mechanical Waves Mechanical Waves There are two basic ways to transmit or move energy from one place to another. First, one can move an object from one location to another via kinetic energy. Second, one can move energy in the form of waves. Waves are energy carriers, moving energy through distances as a result of vibratory motions. Examples of waves are seen or heard around us everyday; ripples on water, sound, sunlight, etc. There are two general classes of waves: electromagnetic waves and mechanical waves. Electromagnetic waves (also known as light) are capable of moving through objects or through the vacuum of space. We will study electromagnetic waves and light in Physics 30. Mechanical waves, on the other hand, require a medium of some kind (i.e. - air, water, a string, a spring, metals, rocks, etc.) through which they can travel. Mechanical waves cannot travel through a vacuum since there is no substance available to support and maintain the wave energy. There are two types or classes of mechanical waves: transverse waves and longitudinal waves. Transverse Waves For a transverse wave the particles in the medium vibrate perpendicular to the direction of motion of the wave energy. A transverse wave has crests (positive amplitude) and troughs (negative amplitude) as shown in the diagram below. Longitudinal Waves For longitudinal waves (also called compression waves and pressure waves) the particles in the medium vibrate parallel to the direction of motion of the wave energy. The wave travels by causing instantaneous compressions and rarefactions in the medium. A compression is a region where the particles are instantaneously more highly concentrated and a rarefaction is a region where the particles are instantaneously further apart. Of course, once the wave passes through a given region, there is no trace of the wave left behind.

2 Wave Pulses and Waves A wave pulse is a disturbance that has a beginning and an end. The word wave or wave train is usually reserved for a continuous series of pulses. The amplitude of a wave pulse or a wave is the maximum displacement of the medium from the equilibrium position. Wavelength, Period, and Frequency For any type or class of wave, the wavelength (λ - pronounced lam-da ) of a wave is equal to the distance (unit is metres) between two points which are in the exact same phase or position on the wave. The period (T) is the time it takes for the wave to travel one wavelength of distance, and the frequency (f) is the number of pulses or vibrations or wavelengths which occur in one second. The unit for period is usually seconds and the unit for frequency is Hertz (Hz). As we saw in circular motion, the relationship between period and frequency is given by:

3 Speed and the Universal Wave Equation All waves move through different materials or media (plural for medium) with different certain speed. The magnitude of the speed will depend on a number of factors. For instance, sound waves travel through rocks faster than they travel through air since the rock has a greater density than air. Another factor is rigidity; the more rigid the medium, the faster the speed of a wave in that medium. For all waves, there is a simple relation between speed, frequency and wavelength. From our previous work we know that speed is given by: For a wave the distance travelled in one cycle is one wavelength (Δd = λ) and the time required for one cycle is the period (Δt = T = 1/f). Thus: This is called the Universal Wave Equation because it applies to all kinds of waves. Example: A series of waves with a wavelength of 2.5 m have a frequency of 50 Hz. What is the speed of the wave series? Example: A wave series with a wavelength of 10 m has a speed of 300 m/s. What is the frequency of the waves?

4 Reflection When a wave or a pulse on a string or rope meets a boundary of some kind, some or all of the wave energy will be reflected by the boundary. The manner in which the wave is reflected depends on the nature of the boundary. If the boundary is a fixed end, the reflected pulse will be inverted relative to the incident pulse. The reflection of a pulse on a string with a fixed end point: As the pulse arrives, it exerts a vertical force on the fixed anchor point, which in turn exerts an equal and opposite force on the string. When the string tugs up, the anchor point tugs down. This downward force on the string generates an upside-down (inverted) reflected pulse.

5 If the boundary involves a free end, the reflected pulse will be upright relative to the incident pulse. Refraction Refraction occurs when a wave changes from one medium to another. When a wave refracts, the speed and wavelength change but the frequency and period remain the same. When a wave pulse moves from one rope to another, some of the wave energy is reflected (partial reflection) and some is transmitted into the next rope. In the examples below, note that the transmitted wave is upright in both cases. However, the orientation of the reflected wave depends on the difference in speed between the two media. For slow to fast reflection, the reflected pulse is upright. For fast to slow reflection, the reflected pulse is inverted. Note that the amplitude of the incident pulse is divided into two smaller amplitudes for the reflected and transmitted pulses. The reflection of a pulse on a string with a free end point: The free end rises until all of the energy of the pulse is stored elastically. It comes to rest at a maximum vertical displacement of twice the height of the crest. Carried by its own inertia the end segment pulls upward on the string, generating a reflected pulse that is right side up.

6 Interference When two waves moving in opposite directions meet they do not reflect off each other, rather they pass through one another. As they do so, the waves superimpose and interfere with each other creating a temporary waveform that is the sum of the two waves. Thus if the waves meet in phase (crest to crest and trough to trough) then the sum of the amplitudes of the waves produces a larger waveform. This is called constructive interference. On the other hand, if the waves meet out of phase (crest to trough and trough to crest), the sum of the amplitudes tends to cancel each other out resulting in destructive interference. Constructive interference: Destructive Interference: Standing Waves The amplitude and wavelength of interfering waves are often different. But waves with the same amplitude and wavelength travelling in opposite directions results in an interesting interference pattern called a standing wave interference pattern or simply a standing wave. In interference pattern 1 below we have two waves of identical amplitude and wavelength travelling in opposite directions. However, since the waves are completely out-of-phase complete destructive interference occurs at every point resulting in zero displacement. For interference pattern 2, the waves are in-phase resulting in complete constructive interference.

7 Practice Problems 1. Determine the wavelengths of the following waves by measuring with a ruler. Check your answers by measuring the wavelength at different locations. 2. Use the following diagram to answer the following questions. (Scale: 1 small square = 10 cm) a. Crests occur at which points? b. Troughs occur at which points? c. Which point is one wavelength away from point C? d. Which point is one-half of a wavelength from point B? e. What is the wavelength? f. What is the amplitude? g. If the frequency is equal to 40 Hz, what is the speed of the wave?

8 3. What is a wave? 4. Draw a transverse wave. Label the wavelength, crest, trough and equilibrium line. 5. Sketch a negative wave pulse of amplitude 2.5 cm. If it is imagined to be travelling from right to left, mark on your diagram the direction of motion of the particles of the medium at the leading and trailing edges of the pulse. 6. What happens to the energy associated with a wave after the wave has been damped out? 7. Sketch a longitudinal wave including wavelength, compressions and rarefactions. 8. For each of the following examples of periodic motion, determine the period and frequency. a. a metronome clicks 50 times in 15 s b. a patient breathes 18 times in 30 s c. a flywheel makes 300 revolutions in one minute d. a cricket chirps 30 times in 15 s

9 9. People can actually use their bodies as particles in a medium to form a wave. In each of the diagrams, determine if the wave is transverse or longitudinal and label the appropriate parts. a. Fans do the wave at a football game. b. People push in the line for football tickets. 10. Match the following terms to the appropriate phrase: Period Frequency Amplitude Displacement Sine Curve Vibration a. the motion of a pendulum b. an S shape on its side c. number of vibrations per second d. the completion of one cycle e. the location of an object f. time to complete one vibration Simple Harmonic Motion g. position of maximum displacement 11. A meter stick is held vertically behind an object on a spring. The object vibrates from the 20 cm mark to the 32 cm mark on the stick. Find: a. the equilibrium position b. the amplitude c. the displacement when the object is at the 20 cm mark, 24 cm mark, 26 cm mark, and 30 cm mark.

10 12. The crest of a wave in a ripple tank travels 60 cm in 2.0 s. If the distance between crests is 0.50 cm, what is the frequency of the wave? (60 Hz) 13. The wave generator in a ripple tank produces circular wave patterns at a frequency of 6.0 Hz. A student measures the distance on the floor between troughs of the 3 rd and 8 th waves to be 7.5 cm. What is the speed of the wave? (9.0 cm/s) 14. Radio telescopes receive waves from distant stars. If such a wave has a wavelength of 21 cm and travels at the speed of light (c = 3.00 x 10 8 m/s), what is the frequency? (1.43 x 10 9 Hz) 15. A tuning fork that vibrates at 256 Hz produces a sound wave that is 130 cm long. a. What is the speed of sound in air? (332.8 m/s) b. A sound wave in a steel rail has a period of x 10-3 s and a wavelength of 10.5 m. What is the speed of sound in steel? (6510 m/s) c. Two men are standing beside a railway line and they are 500 m apart. If one man strikes the rail with a hammer how long does it take for the other man: i. to see the hammer hit the rail? (1.76 x 10-6 s) ii. to hear the sound from the rail line? (7.68 x 10-2 s) iii. to hear the sound in the air? (1.50 s)

11 16. Using the terms frequency, wavelength and speed, fill in the blanks: and are determined by properties of the medium, while is determined by the source of the wave. 17. A wave in a coiled spring travels at 10 cm/s. a. If the frequency of the wave is 2 Hz, what is the wavelength? (5.0 cm) b. If the frequency of the wave is doubled to 4 Hz does the wave speed change? c. What would occur if the frequency were reduced to 1 Hz? 18. The human ear can hear frequencies ranging from 20 Hz to Hz. If the speed of sound in air is 340 m/s, what is the range of wavelengths for audible sound? (17 m, m) 19. If the wavelength of a wave is quadrupled and the speed is quartered, what happens to the frequency? (decreased by 16 times) 20. Would you increase or decrease the frequency of a ripple tank generator to produce a wave with a longer wavelength? Explain. 21. The speed and wavelength of deep water waves are 12 cm/s and 1.5 cm respectively. When the water waves enter a shallow region, their speed is reduced to 8.0 cm/s. What is the wavelength in the shallow region? (1.0 cm) 22. The speed and wavelength of shallow water waves are 12 cm/s and 1.5 cm respectively. When the waves enter a deeper region the wavelength increases to 2.0 cm. What is the speed in the deep region? (16 cm/s)

12 23. A 12 Hz wave travels from deep to shallow water. As it does so, its speed changes from 20 cm/s to 16 cm/s. What are the wavelengths in each region? (1.67 cm, 1.33 cm) 24. In the diagrams below a high speed rope is attached to a low speed chain. Using a diagram and words, explain what happens when (A) a pulse originates on the rope side and (B) when a pulse originates on the chain side. A. B. 25. The pulses shown in the following diagrams are moving toward each other. Sketch the shape of the resulting wave pattern at the moment that they directly overlap.

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