Formula Sheet

RMS

Sine Wave: dBRMS=x dBp2where dBp=decibels peak \text{Sine } \allowbreak \text{Wave: } dB_{RMS} =\frac{x\ dB_{p}}{\sqrt{2}}\text{, } \allowbreak \text{where } dB_{p} =\text{decibels } \allowbreak \text{peak}

Frequency

f=1T , where f=frequency and T=period f=\frac{1}{T}\text{ , } \allowbreak \text{where } f=\text{frequency } \allowbreak \text{and } T=\text{period}

f=νλwhere f=frequency, ν=wave speed, and λ=wavelength f=\frac{\nu }{\lambda }\text{, } \allowbreak \text{where } f=\text{frequency, } \nu =\text{wave } \allowbreak \text{speed, } \allowbreak \text{and } \lambda =\text{wavelength}

Wavelength

λ=νf , where f=frequency, ν=wave speed, and λ=wavelength \lambda =\frac{\nu }{f}\text{ , } \allowbreak \text{where } f=\text{frequency, } \nu =\text{wave } \allowbreak \text{speed, } \allowbreak \text{and } \lambda =\text{wavelength}

Distance

d=rt , where d=distance, r=rate, t=time  d=rt\text{ , } \allowbreak \text{where } d=\text{distance, } r=\text{rate, } t=\text{time } \allowbreak \text{}

Time

t=dr , where d=distance, r=rate, t=time (To get ms from seconds multiply by 1000) t=\frac{d}{r}\text{ , } \allowbreak \text{where } d=\text{distance, } r=\text{rate, } t=\text{time } \allowbreak \text{(To } \allowbreak \text{get } ms\text{ from } seconds\text{ multiply } \allowbreak \text{by } 1000\text{)}

Thresholds

Pascals

20μPa or 20×106 Pa 20\micro Pa\text{ or } 20\times 10^{-6} \ Pa

Dynes

.0002 dynescm2 .0002\ \frac{dynes}{cm^{2}}

Recall since these are the same threshold we can say they are equal thus: \text{Recall } \allowbreak \text{since } \allowbreak \text{these } \allowbreak \text{are } \allowbreak \text{the } \allowbreak \text{same } \allowbreak \text{threshold } \allowbreak \text{we } \allowbreak \text{can } \allowbreak \text{say } \allowbreak \text{they } \allowbreak \text{are } \allowbreak \text{equal } \allowbreak \text{thus:}
.00002 Pa.0002 dynescm2=1=.1Padynescm2 \frac{.00002\ Pa}{.0002\ \frac{dynes}{cm^{2}}} =1=.1\frac{Pa}{\frac{dynes}{cm^{2}}}

Sound Power

1012 Wm2 \frac{10^{-12} \ W}{m^{2}}

deciBells

dBSPL (Sound Pressure Levels)

dBSPL=20log(xmeasuredxref) dB_{SPL} =20\log\left(\frac{x_{measured}}{x_{ref}}\right)

dBSP (Sound Power)

dBSP=10log(xmeasuredxref) dB_{SP} =10\log\left(\frac{x_{measured}}{x_{ref}}\right)

Uncorrelated Sound

Total dB = 10 log(10x1 dB SPL10+10x2 dB SPL10+...+10xn dB SPL10) Total\ dB\ =\ 10\ \log\left( 10^{\frac{x_{1} \ dB\ SPL}{10}} +10^{\frac{x_{2} \ dB\ SPL}{10}} +...+10^{\frac{x_{n} \ dB\ SPL}{10}}\right)

Correlated Sound

Total dB = 20 log(10x1 dB SPL20+10x2 dB SPL20+...+10xn dB SPL20) Total\ dB\ =\ 20\ \log\left( 10^{\frac{x_{1} \ dB\ SPL}{20}} +10^{\frac{x_{2} \ dB\ SPL}{20}} +...+10^{\frac{x_{n} \ dB\ SPL}{20}}\right)

Pressure Total

Total dBSPL=10log(p12+p22+...+pn2p02) Total\ dB_{SPL} =10\log\left(\frac{p_{1}^{2} +p_{2}^{2} +...+p_{n}^{2}}{p_{0}^{2}}\right)

where: p is pressure and p0 is its reference pressure.  \text{where: } p\text{ is } \allowbreak \text{a } \allowbreak \text{pressure } \allowbreak \text{and } p_{0}\text{ is } \allowbreak \text{its } \allowbreak \text{reference } \allowbreak \text{pressure. } \allowbreak \text{}

Sound at a Distance

Total dBSPL=Const dBSPL20log(DobservedDref) Total\ dB_{SPL} =Const\ dB_{SPL} -20\log\left(\frac{D_{observed}}{D_{ref}}\right)

Reflected Sound

Reflection dBSPL at listening position = 20log(Direct Path lengthReflected Path length) Reflection\ dB_{SPL} \ at\ listening\ position\ =\ 20\log\left(\frac{Direct\ Path\ length}{Reflected\ Path\ length}\right)

Assumes inverse square propagation. Note this formula gives the decrease in level. \text{Assumes } \allowbreak \text{inverse } \allowbreak \text{square } \allowbreak \text{propagation. } \allowbreak \text{Note } \allowbreak \text{this } \allowbreak \text{formula } \allowbreak \text{gives } \allowbreak \text{the } \allowbreak \text{decrease } \allowbreak \text{in } \allowbreak \text{level.}

Delay

Reflected delay

Reflection delay = (Reflected path length)  (Direct path length)ν Reflection\ delay\ =\ \frac{( Reflected\ path\ length) \ -\ ( Direct\ path\ length)}{\nu }

Where ν is the speed of sound, assumes completely reflective surface.  \text{Where } \nu \text{ is } \allowbreak \text{the } \allowbreak \text{speed } \allowbreak \text{of } \allowbreak \text{sound, } \allowbreak \text{assumes } \allowbreak \text{a } \allowbreak \text{completely } \allowbreak \text{reflective } \allowbreak \text{surface. } \allowbreak \text{}

Diffusion

Lowest Frequency Diffused

f=11304D f=\frac{1130}{4D}

Where f=frequency, and D=depth \text{Where } f=\text{frequency, } \allowbreak \text{and } D=\text{depth}

Room Modes

For Axial

f=565L or f=ν2D f=\frac{565}{L}\text{ or } f=\frac{\nu }{2D}

Where L=dimension of interest, v=speed of sound, and D=distance between walls \text{Where } L=\text{dimension } \allowbreak \text{of } \allowbreak \text{interest, } v=\text{speed } \allowbreak \text{of } \allowbreak \text{sound, } \allowbreak \text{and } D=\text{distance } \allowbreak \text{between } \allowbreak \text{walls}

General Formula

f=c2p2L2+q2W2+r2H2 f=\frac{c}{2}\sqrt{\frac{p^{2}}{L^{2}} +\frac{q^{2}}{W^{2}} +\frac{r^{2}}{H^{2}}}

Where \text{Where}
p, q and r are room mode numbers (always integers) p,\ q\text{ and } r\text{ are } \allowbreak \text{room } \allowbreak \text{mode } \allowbreak \text{numbers } \allowbreak \text{(always } \allowbreak \text{integers)}
L, W and H are dimensions of the room L,\ W\text{ and } H\text{ are } \allowbreak \text{dimensions } \allowbreak \text{of } \allowbreak \text{the } \allowbreak \text{room}
c is the speed of sound.  c\text{ is } \allowbreak \text{the } \allowbreak \text{speed } \allowbreak \text{of } \allowbreak \text{sound. } \allowbreak \text{}

Absorption

A=Sa A=Sa

Where A=total absorption in Sabines, S=surface area, a=absorption coefficient  \text{Where } A=\text{total } \allowbreak \text{absorption } \allowbreak \text{in } \allowbreak \text{Sabines, } S=\text{surface } \allowbreak \text{area, } a=\text{absorption } \allowbreak \text{coefficient } \allowbreak \text{}

Reverb Time

RT60=V(0.049)Atotal RT_{60} =\frac{V( 0.049)}{A_{total}}

Where V=volume, Atotal=total absorption in Sabines. \text{Where } V=\text{volume, } A_{total} =\text{total } \allowbreak \text{absorption } \allowbreak \text{in } \allowbreak \text{Sabines.}