FREQUENCY, OCTAVE, and WAVELENGTH
Frequency (f) is the number of cycles that the periodic signal completes in one second. The unit Hz (Hertz). The pure tone or the sine wave has a single frequency. The sound and the noise usually are not pure tones. Depending on Fourier theory (the complex signal can be synthesized from sine signals (or pure tones) of different frequencies, different amplitudes and different time delays or phases), we can describe the sound as many pure tones with defined amplitudes (or intensities). We can hear sounds with frequencies 16 Hz to 20,000 Hz. Sounds that are out of this range can not be heard regardless of their intensities. The human hearing system (the ears and related perception system in the brain) is more sensitive to frequencies in the range of 1000 Hz-3000 Hz.
The human hearing system is unable to distinguish between two separate sounds with frequencies too close to each other. In other words, slight change in frequency of the tone will not be audible unless the frequency changing is greater than a defined value. The higher the given frequencies, the wider the frequency of the tone can be changed without audible differences. As a result, we can divide all the audible frequencies into 220 ranges, the higher the frequency the wider the range. Therefore, we need a certain name or unit to describe these ranges. This is exactly the concept of the octave. The interval between two frequencies having a ratio of 2:1. When we need a greater frequency resolution for some studies we can use less value unit like 1/3 octave.
SOUND LEVELS and DECIBEL
These definition are related to the amplitude of the signal. We can describe the absolute value of the sound intensity (I) in Watts per meter square or W/m^2. As we discussed for frequency, the change in the sound intensity (or sound pressure level, SPL) should be more than certain values in order to be noticeable. The full range of the audible sound intensity values can be divided into 100 ranges, the higher the sound intensity (louder the sound) the wider the range. To determine the intensity in regard of this human perception it was good idea to use the Bel: the logarithm of the ratio of two power values. If we have, for example, intensity I=0.1 W/m^2 and we change it to 1 W/m^2, the resulting sound level (note that, we added the word level when we use logarithmic values) will be 1 Bel considering 0.1 W/m^2 as the reference value (to take the log). Hence the Bel unit is considerably large, we use one tenth of it(the same concept as we discussed earlier for more frequency resolution), and the resulting unit we call decibel or (dB). In sound application the reference value is 20mðRða for sound pressure or 10E-12 W/m^2 for sound intensity. These values give the 0dB sound level, and represent the human threshold of hearing (the lowest level that can be perceived). The threshold of pain or feeling (the level which causes pain in the ears) is about 120 dB.
SOUND ABSORPTION COEFFICIENT
This coefficient describes the efficiency of the material or the surface to absorb the sound. The ratio between the absorbed sound energy and the incident energy is the sound absorption coefficient. The sound absorbing materials and constructions can be divided, for architectural purposes, into 4 types depending on the way the absorption is mainly performed: 1-turning the sound energy into heat, like fiberglass or carpet. 2-vibrating with a specific frequency when the sound hits the surface, like lightweight panels or 5/8" gypsum board. (These materials effectively absorb the sound only at certain frequencies usually with some kind of distortions in the resulting sound) 3-turning the sound energy into heat in the neck of the cavities (Helmholtz resonator) like sound blocks. (This construction has good absorption at low frequencies) 4-allowing the sound to go through like some types of grid system or lay-in ceiling with sound leakage about it.
The most common way to measure sound absorption coefficient is to lay the piece of the material in the reverberation room (a room which has very long Reverberation Time, or RT), then measure the RT so the coefficient can determine this procedure. The value of the coefficient for the same material varies with the type of the mounting in the test room. Mounting types are frequently given in acoustical panel manufacturers data sheets.
Diffraction is the change in the direction of propagation of sound waves passing the edge of the obstacle. Diffraction phenomenon significantly depends on the relationship between the wavelength of the sound and the size of the obstacle. The longer the wavelength the stronger is the sound diffraction. This effect happens also to the sound transmitted through openings.
PHYSICAL and GEOMETRICAL (RAY) ACOUSTICS
The sound behavior in the room is strongly effected by the ratio of the frequency (or the wavelength) of the sound to the size of the room. As a result, the audible spectrum can be divided into four regions.
Modes are the resonant frequencies on which the waves interface and form maximums and minimums of sound pressure at certain points of the room. This happens, at low frequencies in comparison with the dimension of the room. The calculation of modes in a rectangular enclosure is simple. The modes become complex and sometimes unpredictable in rooms of different shapes. There are 3 types of modes: axial (2 parallel surfaces contributed to the generation of the modes), tangential (4 surfaces) and the oblique mode (6 surfaces).
The lower frequency of all modes is axial, and it can be calculated as f=c/2L, where c is the speed of sound and L is the room length.
It is important to note that, the modes of the enclosure are weak (in pressure amplitude) when the walls are sound absorbing. To splay the walls (in practice rooms for example) makes modes unpredictable and less organized which, sometimes, can weaken the well defined structure of the maximum and minimum values of the sound pressure.
SOUND DIFFUSION and DIFFUSERS
Sound in an enclosure can described as diffused if the intensity of the sound energy is equal in every location of the room, or the sound energy flows equally in every direction. Many different factors can enhance the diffused sound. These include geometrical irregularities, absent of focusing surfaces, the distribution of absorptive and reflective elements randomly scattered through the space, and the existence of diffusing objects (furniture) or panels (diffusers).
Diffusing panels scatter the sound in all, or in certain directions depending on their type and geometrical dimension. A new type of diffuser is the Schroeder diffuser (Quadratic-residue diffusers). Its diffusion characteristics do not depend solely on its geometrical dimensions but also on an array of wells with depths determined by a listed quadratic residue sequence.
REVERBERATION TIME (RT)
Reverberation time is the time required for the sound level in the room to decay 60dB, or in other words, it is the time needed for a loud sound to be inaudible after turning off the sound source.
The calculation of reverberation time using the Sabin equation assumes that the sound in the room be diffused. In practice, RT equations are close enough to describe the sound build up and attenuation in the room. In the case where the sound in the room is not diffused enough, like rooms with good absorption surfaces in certain areas, or with an unusual shape (long and narrow, very low ceiling, or many different focusing surfaces), the RT calculation is not accurate. There is a Fitzroy equation to correct the RT calculation for rooms with good absorptive surfaces on a certain axis of the room. The optimum reverberation times for different rooms depends on the volume of the space, the type of the room, and the frequency of the sound. In general terms, the optimum RT for rooms with speech programs is less then the optimum RT for rooms with music performance.
Help you discover how much fun and useful math is. The Acoustics House includes sample problems and solutions typical of those professional designers and builders encounter in their work. Whether you are planning a real or fantasy building project, or using the Acoustics House as an educational tool, this curriculum can help students learn about design and math.
Designing and building a model teaches important math concept and sharpens practical skill from basic arithmetic, measurement and scale to geometry, trigonometry and spatial relations. The predominant skills developed and nurtured while working with complex problem solving and precision are the same skills which are crucial to success at all levels of mathematics and on the job.
To design a house, you have to constantly analyze many things at the same time, establish priorities, and try various approaches to each problem, strategies also important to math and work. For example, if a wall is moved, it generally has multiple consequences. Moving one wall not only changes the area and proportion of the room you are working on, but it changes the space on the other side of the wall as well. In a chain of events, you start by moving one wall, then have to move other walls in other rooms to keep their proper size and shape. Next, the windows and doors have to be moved to compensate for the wall changes, which affect the elevations and possibly the layout of the other floors and roof. These complex, interdependent relationships are what makes house design so challenging and interesting.
DESIGNER AND BUILDER PROBLEMS (Appendix)
Cut and assemble the house on graph paper. Design your own arrangement of windows and doors. These can be cut out of the walls or you can arrange the window and door by cut them out of graph paper. Sketch a floor plan and then organize the furniture, stairs, fireplace and other features to plan your own interior design.
Design a geometric floor pattern for one room. Use the Appendix as a guide to make a list of materials needed for your design. Calculate the area of the walls and ceiling in each room. If one gallon of paint cover 400 square feet, how much paint is needed to cover the walls and ceilings in each room and the entire house? Now, determine how many 4-ft. by 8-ft. sheets of wood paneling are needed to cover the walls. You can sketch the panels on graph to determine the most economical layout for the panels. Calculate the ratio of the window area to floor area habitable room including bedrooms, living room and den. For new single-family home construction, it is recommended that the window area be at least 8% of the floor area. This is necessary for adequate natural light. Determine the volume of each room. To do this multiply the length times the width time, the height of each room. The volume is needed to determine the size of the heating and air conditioning systems.
Calculate the volume of concrete needed for a 4-inch thick floor slab for one room or your entire house. After determining how many cubic feet of concrete are needed, convert your answer into cubic yards, which is how builders order concrete.
Measure and draw the angle of the roof on your home. Designers and builders refer to the slope of a roof by its vertical rise in inches for each horizontal foot. For example, a roof which rises 6-inches for each horizontal foot has a 6/12 pitch. To draw a roof with a 6/12 pitch, use graph paper, make a right triangle with a horizontal length of 12 grid boxes and a height of 6 grid boxes; then use a straightedge to draw the roof slope by connecting the ends of the lines.
Allow students to solve solve math problems in their own way. Encourage students to use graphics to solve math problems. Make students aware that there is no right way to solve a problem. Don't give answers, instead ask questions that make students think. Encourage teamwork.
Noise paths in a building, air path through opening, air-borne noise, solid-borne noise.
The human hearing system has different sensitivities at different frequencies. This means that the perception of noise is not equal at all frequencies. Noise with significant measure levels at high or low frequencies will not be as annoying as it would be when its energy is basically in the middle frequencies. In other words, the measured noise levels will not reflect the actual human reception about the loudness of the noise.
Community noises constantly change in their levels and duration. It can reach 50 dB changes in short time.
Noises in buildings are more stable (over time) than outside community noise. The maximum acceptable background noise level generated by mechanical systems in a building is usually specified in terms of averages A-weighted sound levels, NC, RC, or NCB. The noise criteria (NC) values are determined from the measurements of the octave-band sound levels in an occupied room when the air-conditioning system is on. Then we compare the measured value to standard NC curves. The room criterion (RC) is mostly used for acoustical design of HVAC systems. The measurement values should be taken in an unoccupied room (you can have students calculate echoes Appendix 3).