Introduction to Acoustics Exercises

Transcription

1 Introduction to Acoustics Exercises 1 Fundamentals of acoustics 1. Show the effect of temperature on acoustic pressure. Hint: use the equation of state and the equation of state at equilibrium. 2. Derive the equation of state for p but by including the second term in the Taylor s series expansion. Assuming γ = 1.4, ρ 0 = 1.2 and ρ = 10 4, how much smaller is the second term? 3. The density at equilibrium is increased from ρ 0 to ρ 1 at time t = 0. Derive the equation of mass conservation for this case. 4. Assume the pressure and particle displacement are given by p = Ae j(ω 0t kx) and l = Be j(ω 0t kx), and derive an expression for the amplitude of the displacement of the acoustic particles as a function of frequency. Hint: use the equation of momentum conservation. What is the displacement at 1000Hz (given ω = 2π1000, k = ω 0 /c and ρ 0 = 1.2) for pressure amplitude of 10 Pa? 2 Sound propagation 1. By representing the pressure as p = p(x)e jωt, derive the wave equation for p(x). This is called the Helmholtz equation and shows the spatial behavior of the sound field. 2. For a wave composed of two plane waves traveling in the positive x-direction but with different frequencies, show that the specific acoustic impedance is ρ 0 c. 3. Find the specific acoustic impedance for a standing wave composed of two plane waves propagating along the x-axis in opposite directions, with opposite amplitudes (A and A). Are the pressure and particle velocity in phase? 4. Calculate the intensity of the standing wave described above, and explain your results. 5. Average speech one meter away from the speaker is about 65 db SPL. What is the pressure at this position? 22 March

2 3 Additional exercises for topics 1 and 2 1. Derive expressions for the radial particle velocity and specific acoustic impedance of spherical waves. How does the impedance behaves for very small and very large kr? 2. Derive an expression for the sound intensity of spherical waves, and show that the sound power is independent of the radius. 3. Sound is transmitted in air though a layer with impedance z 2 and thickness L. Show that if z 2 is much larger then the impedance of air, and if the layer is a thin partition such that L 0, then the partition is acoustically transparent (T I = 1). 4. Given a layer of impedance z in air as in question 3, but with a thickness L which is not negligible, at what frequencies will the layer still be acoustically transparent? 4 Sound sources 1. (Q7.1.5C) (a) Show that in the limit ka 1 the specific acoustic impedance at the surface of a pulsating sphere can be approximated by z ρ 0 cka(j + ka). (b) Plot the resistance, reactance and magnitude of the impedance as a function of ka and compare with the impedance calculated without the approximation. (c) Find the values of ka within which the errors in the approximations are less then 10%. 2. Plot the magnitude of the on-axis pressure radiated form a circular piston for a = 0.2λ and a = 2λ, at a frequency of 100Hz. Explain the results and any difference between the two cases. 3. (Q7.1.1) A pulsating sphere of radius a = 0.1m radiates spherical waves into air at 100Hz and with an intensity of 50mW/m 2 at a distance of 1m from the center of the sphere. (a) What is the radiated acoustic power? (b) Calculate the intensity and the amplitude of the sound pressure and particle velocity at the sphere surface, r = a. 5 1D sound - waveguides and filters 1. Given a tube of area S, driven by a piston of velocity U at x = 0, and terminated by an acoustic impedance Z L at x = L. Derive an expression for the acoustic impedance near the piston (x = 0), as a function of the termination impedance. Use the expression to calculate the pressure at x = 0 for a rigid termination Z L and compare to the result derived in class directly for this case. 2. You are given the task to design an acoustic filter that will reduce the noise from the outlet of an air conditioning duct. You have three options: (a) introduce a volume within the duct by increasing its area within a selected section, (b) reduce the duct areas within a selected section, (c) introduce a Helmholts resonator. Present 22 March

3 the advantages and disadvantages of each solution. For what noise type will each solution be used? 6 3D sound - room acoustics 1. Write a Matlab program that calculates the modal frequencies up to 500 Hz in a room of dimensions 10m 5m 3m. Plot a histogram of these frequencies as a function of frequency and estimate the increase in density as a function of f. 2. For the room in the previous problem where would you place a loudspeaker, such that no (or very little) sound level is produced in the room at 68.9 Hz? 3. A room 10m 10m 4m has an average absorptivity of ā = 0.1. (a) Calculate the reverberation time. (b) What must be the power output of a source if it is to produce a steady-state sound pressure level of 65 db SPL? (c) At what rate in W/m 2 is sound energy incident on the walls of the room? 4. For the previous problem, what will be the rate that energy grows in the room if there is no absorption? 5. A source generating sound at 68.9 Hz is placed in the room described in the first question. Assume some small absorption greater then zero, can the reverberation time at this frequency can be calculated using the diffuse field model? Explain. 7 Hearing 1. A white background noise of overall level of 63 db SPL and bandwidth of Hz is played together with a tone at 2000 Hz of some unknown level. What would be the level of the tone if it is to be heard clearly, such that its level is 10 db above that of the noise within the critical band? 2. A noise similar to the one in the previous question is played softly at an overall level of 35 db SPL. If the frequency components of the noise are weighted according to their relative loudness (with 1 khz having a weight of unity), then the overall level is reduced to 30 db (weighted SPL). The noise is then played at a high level of 95 db SPL. What would you expect the new weighted level to be? 3. You are listening to two tones, one at 400 Hz and one at 2000 Hz, both at 80 db SPL. The level of the 2000 Hz tone is reduced until it cannot be heard any more. In your case this level was found to be 40 db SPL. This experiment is repeated, but this time the level of the 400 Hz tone is reduced. What do you expect the threshold level of the 400 Hz tone in this case to be (the level where it is just heard): (a) 40 db SPL (b) 80 db SPL (c) its threshold is unaffected by the 2000 Hz tone (d) there is no way to know. 22 March

4 8 Loudspeakers 1. You are provided with a moving-coil loudspeaker with given parameters and response, and are asked to increase its membrane area by two. You are asked to provide your supervisors with a short report explaining what will be the expected effect of this change. 2. You are required to design a loudspeaker with as high efficiency as possible. If you were able to control R, R m, R r, the electrical, mechanical and acoustical resistances, how would you choose them to achieve your task? 3. A loudspeaker has a mechanical resonance frequency of 100 Hz and a membrane mass of 10 grams and radius 10 cm. You are asked to design a small cabinet for the loudspeaker, but which does not affect its low frequency performance. What would be the upper limit of the volume of this cabinet? 9 Microphones 1. One of the most important features of a microphone is its frequency response, i.e. the high and low cut-off frequency and the sensitivity. You are asked to design a microphone with high sensitivity and a broad frequency response - how would the various microphone parameters affect your task? 2. You are operating a computer in a noisy environment and requires to use a microphone on your desktop for communications. If the noise is arriving from a distant source in a known direction in front of you, and assuming reflections are negligible in you environment, would you use a Cardiod or a close-talk microphone? Explain. 3. You are given two bidirectional, or figure-of-eight microphones, one with the nulls towards the x-axis, and the other with the nulls towards the y-axis. You are also given an omni-directional (pressure) microphone. Show how using these three microphones you can design a cardiod microphone which can be electronically steered to any direction on the x-y plane. 10 Active noise reducing headset 1. Passive noise reducing headsets might not always be completely air-tight, and so a small leak might occur between the cushion and the head. Assume such a leak can be modeled as an acoustic resistance, and derive the passive attenuation equation which includes the leak. What frequencies will be affected the most? 2. Following the previous question, explain why headphones which are not fitted properly to the head will loose their bass. 22 March

5 3. Your task is to improve the sensitivity of audio headphones which cover the ear and form a tight volume around the ear. Which of the following do you think will help: (a) decreasing the mass of the loudspeaker membrane, (b) decreasing the stiffness of the loudspeaker membrane, (c) decreasing the volume enclosed by the headphones, (d) increasing the volume enclosed by the headphones. 22 March