Name: Waves & Sound Hr: Vocabulary Wave: A disturbance in a medium. In this chapter you will be working with waves that are periodic or that repeat in a regular pattern. Wave speed = (wavelength)(frequency) OR v = λ f The SI unit for wave speed is the meter per second (m/s). The speed of sound in air increases with air temperature. For the following exercises, the speed of sound will be written as 340.0 m/s. All electromagnetic radiation including radio waves and light waves travel at the speed of light, 3.0 x 10 8 m/s. The wavelength of a wave is the distance from one point on a wave to the next identical point on the same wave, for example, from crest to crest or trough to trough. The symbol for wavelength is the Greek letter lambda, λ. The SI unit for wavelength is the meter (m), which is the same unit used for length in earlier chapters. The SI unit for frequency is the hertz (Hz), or (1/second). When talking about the broadcast frequency of a radio station, frequencies of FM radio stations are given in megahertz, or MHz, and frequencies of AM radio stations are given in kilohertz, or khz. 1 MHz = 1 x 10 6 Hz and 1 khz = 1 x 10 3 Hz Solves Examples Example 1: Radio station WBOS in Boston broadcasts at a frequency of 92.9 MHz. What is the wavelength of the radio waves emitted by WBOS? Given: v = 3.00 x 10 8 m/s Unknown: λ =? f = 92.9 x 10 6 Hz Original equation: v = λ f Solve: λ= v f = 3.0 x 108 m/s 92.9 x 10 6 Hz = 3.23 m Therefore, the distance from one point on the wave to the next identical point on the same wave is 3.23 m. Example 2: In California, Clay is surfing on a wave that propels him toward the beach with a speed of 5.0 m/s. The wave crests are each 20. m apart. a) What is the frequency of the water wave? b) What is the period? a. Given: v = 5.0 m/s Unknown: f =? λ = 20.0 m/s Original Equation: v = λ f Solve: f= v λ 5.0 m/s = = 0.25 Hz 20. m b. Given: f = 0.25 Hz Unknown: T =? Original Equation: T= 1 f
Solve: T = 1 = 1 = 4. 0 s f 0.25 Hz One crest comes along every 4.0 seconds. Standing Waves Waves in Strings When a string is plucked, a wave will reflect back and forth from one end of the string to the other, creating nodes (points of minimum movement) and antinodes (points of maximum movement). This is called a standing wave because it appears to stand still. If there is only one antinode, as shown in the diagram, the frequency is called the fundamental frequency. It is the lowest frequency of a vibrating string that is fixed at both ends. Multiples of the fundamental frequency are called overtones. Fundamental Waves in Pipes Waves in pipes that are open at both ends behave much like waves in strings, while waves in pipes that are closed in one end behave a little differently. It is important to remember than antinodes always form at the open end of pipes while nodes form at the closed end. Beats If two different frequencies sound simultaneously, the wavelengths will differ, and the crests and troughs of each wave will overlap in a way that causes variations in loudness. There will be moments of reinforcement and
moments of cancellation as the wave patterns interact. The resulting sound is a series of beats, which occur when the wave sum reaches its greatest amplitude. The beat frequency can be found by taking the absolute value of the difference between the two frequencies of the interacting waves. f beat = f 1 -f 2 Solved Examples Example 3: An orchestra tunes up for the big concert by playing an A, which resounds with a fundamental frequency of 440. Hz. Find the first and second overtones of this note. The first overtone is 2 times the fundamental frequency: f 2 = 2f o so f 2=2(440. Hz) = 880. Hz The second overtone is 3 times the fundamental frequency: f 3 = 3f o so f 3=3(440. Hz) = 1320 Hz Example 4: Zeke plucks a C on his guitar string, which vibrates with a fundamental frequency of 261 Hz. The wave travels down the string with a speed of 400. m/s. a) What is the length of the guitar string? b) Would Zeke need longer or shorter strings to play the fundamental frequency for higher notes? a. Given: fundamental (1 st harmonic) Unknown: L =? f = 261 Hz Original Equations: v = λ f v = 400 m/s L = ½ λ Solve: λ= v (400 m/s) = =1.53 m f (261 Hz) L= 1 λ= 1 (1.53 m)=0.766 m 2 2 b. If the wave speed remains the same for each string, as f gets larger, L gets smaller. Therefore, the higher the note, the shorter the string required to hear the fundamental frequency. Example 5: In his physics lab, Sanjiv finds that he can take a long glass tube and fill it with water, using the air space at the top to simulate a pipe closed at one end. If Sanjiv holds a tuning fork, which vibrates with a fundamental frequency of 440 Hz, over the mouth of the pipe, how long is the air column if it vibrates at the same frequency? Given: fundamental (1 st harmonic) Unknown: L =? f = 440 Hz Original Equations: v = λ f v = 340 m/s L = ¼ λ Solve: λ= v (340 m/s) = = 0.77 m f (440 Hz) L= 1 λ= 1 (0.77 m)=0.19 m 4 4
Grading: Show all work, including formulas, algebra, plugged-in numbers, units, & circled answers. INCLUDE A DIAGRAM FOR ALL STANDING WAVE PROBLEMS. Use v=340 m/s for the speed of sound in air. 1. If a Wave Pool wave generator produces 10 meter-long waves every 4 seconds, a) calculate the wave speed. b) If the period of the generated waves was decreased but the wavelength remained the same, what would happen to the wave speed? 2. Dogs are capable of hearing much higher frequencies than humans. A dog whistle, silent to humans, creates a sound wave with a frequency of 3.5 x 10 4 Hz. What is the wavelength of this sound wave? 3. a) What is the frequency of a 0.44 m long sound wavelength produced by a soprano singer? b) What is the period of this sound wave? c) Would a shorter wavelength produce a higher or lower pitch? Why? 4. At a music festival in New Mexico, the Oak Ridge Boys are playing at the end of a crowded 175-m field when Ronny Fairchild hits a note on the keyboard that has a frequency of 382. Hz. a) How many wavelengths are there between the stage and the last row of the crowd? b) How much delay is there between the time a note is played and the time it is heard in the last row?
5. Before a concert, a guitarist and bassist tune their instruments to match pitch. At first, the guitar produces a frequency of 80 Hz and the bass produces a frequency of 84 Hz. a) What beat frequency is created? b) Should the guitarist tighten or loosen his string to match pitch with the bassist? Why? 6. The violin A string (440 Hz) has a length of 30 cm. a) At what speed does the fundamental frequency travel along the string? b) How is the standing wave that produces the fundamental frequency created on the violin string? 7. The fundamental frequency of a bass violin string is 928 Hz and occurs when the string is 0.82 m long. How far from the lower fixed end of the bass violin should you place your fingers to allow the string to vibrate at a fundamental frequency 3 times as great?
8. A trumpet is an open tube with a length of 150 cm. a) Calculate the lowest note (fundamental frequency) that it can play. b) Calculate the frequency of the trumpet s 1 st overtone (2 nd harmonic). 9. Aaron blows across the opening of a partially filled 18.0 cm high soft drink bottle and finds that the air vibrates with a fundamental frequency of 512 Hz. How high is the liquid in the bottle? 10. If a clarinet (modeled as a closed tube) produces a 2 nd overtone (5 th harmonic) with a frequency of 739 Hz, a) what is the length of the clarinet? b) If the clarinet player holds more buttons down, increasing the tube length, will the pitch increase or decrease? Why?