9.1 Questions 1. Describe how waves combine. k/u (a) What happens when waves that are in phase combine? (b) What happens when waves that are out of phase combine? 2. Use the principle of superposition to determine the resulting waveform when the waves in Figure 12 interfere with each other. T/ C (a) (b) (C) Figure 12Mach Number Ernst Mach (1838-1916), an Austrian physicist, researched sound waves and devised a way to describe air speeds of objects in terms of the speed of sound. Mach's approach relates the local speed of sound and the airspeed (speed relative to the surrounding air) of an object, such as an aircraft. (Section 8.6 describes what happens when aircraft travel at the speed of sound, as well as the history of aircraft attempting to reach that Mach number (M) the ratio of the speed.) The ratio of the airspeed to the local speed of sound is called the Mach number: airspeed of an object to the local speed of sound M = airspeed of object local speed of sound Note that the ratio has no units. For this reason, when describing the speed of an object using the Mach number, we say Mach 1, Mach 2, and so on. Mach 1, for instance, means that the object is travelling at the speed of sound. An aircraft's Mach number is not fixed-it changes depending on the speed of sound in its vicinity. For example, an aircraft travelling at fixed speed from Helsinki, Finland, to Havana, Cuba, would have different Mach numbers at the two places because of different air temperatures and pressures. Tutorial 2 shows how to calculate Mach numbers. Tutorial 2 Calculating the Mach Number Sample Problem 1 An aircraft is flying at 905 km/h in air at the temperature -50.0 C. Calculate the Mach number associated with this speed. Given: T = -50.0 .C; v = 905 km/h Required: M Analysis: Calculate the speed of sound in air at a temperature of -50.0 .C. The speed of the airplane must be converted into metres per second and the calculation made using the Mach number equation. Solution: V = 331.4 m/s + (0.606 m/s/C) T = 331.4 m/s + (0.606 m/s/C) (-50.0 .C) v = 301.1 m/s (one extra digit carried) 905 km 1000 m 1 h - = 251.4 m/s 1 km 3600 s airspeed of object M = . local speed of sound 251.4 m/s 301.1 m/s M = 0.835 Statement: The Mach number is 0.835. Practice 1. If the local speed of sound is 344 m/s and an aircraft is flying at 910 km/h, what is the Mach number? Till [ans: 0.73] 2. If the Mach number is 0.93 and the local speed of sound is 320 m/s, what is the speed of an airplane in these conditions? T/ [ans: 3.0 x 102 m/s = 1100 km/h] 3. If the Mach number is 0.81 and the speed of an airplane measured by radar is 850 km/h, what is the local speed of sound in kilometres per hour? r [ans: 290 m/s = 1.0 x 10 km/h] 394 Chapter 8 . Vibrations and Waves NELSPH3U: Unit 4 A2: Sound Waves Practice Worksheet 1. If the temperature of the air in your region is 32 C, what is the speed of sound in air at that temperature? (351 m/s) 2. If the speed of sound near you is 333 m/s, what is the ambient temperature? (2.64 .C) 3. If the speed of sound near you is 350 m/s, what is the ambient temperature? (31 .C) 4. If the local speed of sound is 344 m/s and an aircraft is flying at 910 km/h, what is the Mach number? (0.73) 5. If the Mach number is 0.93 and the local speed of sound is 320 m/s, what is the speed of an airplane in these conditions? (3.0x102 m/s = 1100 km/h) 6. If the Mach number is 0.81 and the speed of an airplane measured by radar is 850 km/h, what is the local speed of sound in kilometers per hour? (290 m/s = 1.0x103 km/h)Tutorial 1 Standing Waves Sample Problem 1 Solution: Calculate the wavelength: The speed of a wave on a string with a fixed end and a free end is 350 m/s. The frequency of the wave is 200.0 Hz. What length f of string is necessary to produce a standing wave with the first 350 m/s harmonic? 200.0 Hz Given: free and fixed ends; v = 350 m/s; f = 200.0 Hz; n = 1 A = 1.75 m (one extra digit carried) Required: L Calculate the length of string: Analysis: We first determine the wavelength using the universal L = (2n - 1) wave equation. We can then use the formula for fixed and free 4 ends to calculate the required length. A = ; L = (2n - 1) L = - A 4 = -(1.75 m) L1 = 0.44 m Statement: A string that is 0.44 m long will produce a standing wave with the first harmonic. Sample Problem 2 Solution: 46 = nA 2 The sixth harmonic of a 65 cm guitar string is heard. If the speed of sound in the string is 206 m/s, what is the frequency of the 246 1 = standing wave? n Given: two fixed ends; 46 = 0.65 m; n = 6; v = 206 m/s (2) (0.65 m) Required: f6 6 Analysis: First, we determine the wavelength of the guitar A = 0.2167 m (two extra digits carried) string using the relationship for fixed ends. Then we use the v = fox universal wave equation to calculate the frequency of the standing wave. f6 = Ln =- 206 m/s 0.2167 m 6 = 950 Hz Statement: The frequency of the standing wave is 950 Hz. Practice 1. A 0.44 m length of rope has one fixed end and one free end. A wave moves along the rope at the speed 350 m/s with a frequency of 200.0 Hz at n = 1. T (a) What is L, if the frequency is doubled? [ans: L, = 0.22 m] (b) What is the length of the string if n = 3? [ans: 2.2 m] (c) What is Z, if the speed of the wave on the string is reduced to 200 m/s? [ans: 0.25 m] 2. The speed of a wave travelling along a 0.65 m guitar string is 206 m/s. At n = 6, the frequency is 950 Hz. T/ (a) What is the frequency if a string with a wave speed of 150 m/s is used? [ans: 690 Hz] (b) What is the frequency if the string is tightened to make the wave speed 350 m/s? [ans: 1600 Hz]The Speed of Sound Investigation 8.5.1 It has been determined experimentally that the speed of sound through air depends Measuring the Speed of Sound on the density of the air and its temperature. This value increases by 0.606 m/s for (p. 404) every increase of 1 C. Using the equation below, you can calculate the speed of sound In this investigation, you will predict in air at different temperatures: the speed of sound on a certain day and then measure it. Make sure you V = 331.4 m/s + (0.606 m/s/C) T understand how the equation for calculating the speed of sound works. where T is the temperature in degrees Celsius. Tutorial 1 shows how to calculate the speed of sound in air using this equation. Tutorial 1 Calculating the Speed of Sound Sample Problem 1: Determining the Speed of Sound at a Certain Temperature The temperature outside is 23 C. What is the speed of sound in air at this temperature? Given: T = 23 .C Required: v Analysis: The information given can be directly substituted into the equation for the speed of sound in air: v = 331.4 m/s + (0.606 m/s/C) T Solution: V = 331.4 m/s + (0.606 m/s/C) T = 331.4 m/s + (0.606 m/s/C) (23 .C) V = 345 m/S Statement: The speed of sound in air at 23 .C is 345 m/s. Sample Problem 2: Determining Air Temperature If the speed of sound is measured to be 318 m/s, what is the current air temperature? Given: v = 318 m/s Required: T Analysis: To solve for the temperature of the air, rearrange the original equation. Then substitute the given variable and calculate the answer: v = 331.4 m/s + (0.606 m/s/C) T Solution: V = 331.4 m/s + (0.606 m/s/C) T T = V - 331.4 m/s 0.606 m/s/C 318 m/s - 331.4 m/s 0.606 m/s/ C T = -22.1 .C Statement: The temperature of the air is -22.1 .C. Practice 1. If the temperature of the air in your region is 32 C, what is the speed of sound in air at that temperature? To [ans: 351 m/s] 2. If the speed of sound near you is 333 m/s, what is the ambient temperature? [ans: 2.64 .C] 3. If the speed of sound near you is 350 m/s, what is the ambient temperature? T [ans: 31 .C]2. Identify from your own experience an example of a wave that encounters a media boundary. A (a) From your observations, is the amplitude of the reflected wave or transmitted wave increased? (b) Does the change in the medium support your answer to (a)? Explain. 3. Describe from your own experience an example of a free-end reflection and a fixed-end reflection. A 4. Describe the conditions required to form a standing wave. K/U 5. A string is 2.4 m long, and the speed of sound along this string is 450 m/s. Calculate the frequency of the wave that2. An aircraft is flying at Mach 2. What does this mean? K/U 3. An aircraft is travelling at Mach 0.83 in air at 10 C. What is its speed in kilometres per hour? T/ C 4. Explain why the speed of sound varies in the different materials in Table 1 on page 395. K/U C 5. In your own words, define (a) sound intensity, (b) loudness, and (c) decibel. K/U C