SABINE'S REVERBERATION FORMULA

Reverberation Time is the time it takes the sound to decay away 60 dBs.

Outline

• Flutter echoes and other problems
• Typical reverberation times
• Sabine's formula

Next Time: Harmonics

Things to Avoid:

Flutter echo: A common problem occurs when there is distinct path of reflection off two hard walls. One hears fast, successive echoes, or "flutter echoes."

Two distinct decay times: This occurs when there are two distinct chambers, or two distinct paths of reflection, which decay on two fairly different time scales.

Reverberation Time is the time it takes the sound to decay away 60 dBs.

Typical reverberation times:

Practice room: V = 27 m³ , T = 0.6 s ( 3m x 3m x 3m)

Rehearsal room: V = 600 m³ , T = 1.4 s ( 6m x 10m x 10 m)

Large concert hall: V = 20,000 m³ , T = 2.2 s (20m x 32m x 32m)

Reverberation times should be consistent with visual cues, which tell us what the size of the room is. A desirable reverberation time also depends on the type of music. Chamber music (a string quartet, for example) benefits from a shorter reverberation time than orchestral music. See Figure 23.7

It is also important that the lower frequencies have a longer reverberation time. This is what gives warmth to the sound. See Figure 23.8

Sabine's formula

T = 0.16 V / A

where T is the reverberation time, V is the volume of the room and A is the surface area of absorbing material.

This formula is fairly general and we will talk shortly about how to determine A in this equation. But, for now let's assume that the walls in our room are completely absorbing. That is, any sound that hits the wall is absorbed and there is no reflected sound. We can then exactly derive Sabine's formula.

Let's say we play a steady tone for a long time in our room the sound energy will build up and we will call the energy density: e. We will assume the energy density is constant throughout the room

e = sound energy density Joules/m³

The power (Joules/sec) being dissipated in the walls will be the area of the walls times the intensity (Watts/m2) of the sound hitting the walls.

P = A I

The power lost to the walls must equal the time rate of change of the energy.

d/dt (e V) = - P = - AI

V is constant and the intensity equals the energy density times the speed of sound divided by 4. We divide by 4 because sound is going out in all directions, left, right, forward, backward. We do not divide by 6. Why not? Because 4 gives a better answer (closer to what is actually measured). The physical reason is that sound is not really radiating uniformly in all directions so the number must be something less than 6. Sound tends to reflect in a planar surface, which is consistent with using 4 instead of 6.

de/dt V = - A e/4 v

de/dt = - Av/(4V) e

e = e₀ exp ( - Av/(6V) t )

ln (e/e₀) = - A v / (6 V) T

T = -ln (10^-6) x 4 / 344 V / A

= 13.8 x 4 / 344 V/A = 0.161 V/A

Absorption Coefficients

In reality, surfaces do not purely absorb or purely reflect.

The absorption coefficient is the amount of power absorbed divided by the amount of incident power.

The way we estimate the absorbing area, A, in Sabine's formula is by weighting absorbing areas by their absorption coefficient.

A = C₁ x A₁ + C₂ x A₂ + C₃ x A₃ + …

V = the total volume of the room in cubic meters

A_i = surface area

C_i = absorption coefficients

A is the total "fully-absorbing" surface area of the room. Each surface is weighted by its absorption coefficient. You add up all surfaces. Each surface contributes its surface area times its absorption coefficient.

These calculations can become quite involved. You have to take into account windows, doors, hallways, the audience, empty seating, etc. Every surface has a different absorption coefficient.

C is the fraction of energy lost each time the sound wave reflects off the surface. The value of A ranges from 0-1. 0.01 would be highly reflective, 0.3 is fairly absorbing.

Sabine's formula is approximate. However, it works reasonably well if the average absorption coefficient is less than 0.15. That is,

A / A_total < 0.15

It also gives a way to compare various options when choosing size, shape and materials. There are computer programs which do "ray tracing." That is, the computer simulates the motion of the wave fronts and how they reflect off the surfaces and how they are absorbed (CATT Acoustic is the program I am familiar with). These programs also have "diffusion." That is, a way to mimic how sound diffuses by scattering off of complex objects. Ray tracing is also used to simulate how light reflects off objects for three-dimensional visualization and animation. Another common practice is to make a scale model of the concert hall and test the model with higher frequency sound sources so that the wavelengths are in proportion to the size of the model. This was not done for Philharmonic Hall!

SYLLABUS FOR NET

CSIR-UGC National Eligibility Test (NET)

for

Junior Research Fellowship

and

Lecturer-ship

SYLLABUS FORPHYSICAL SCIENCES

PAPER I AND PAPER II

The full Syllabus for Part B of Paper I and Part B of Paper II.

The syllabus for Part A of Paper II comprises Sections I-VI.

I. Mathematical Methods of Physics

Dimensional analysis; Vector algebra and vector calculus; Linear algebra, matrices, Cayley Hamilton theorem, eigenvalue problems; Linear differential equations; Special functions (Hermite, Bessel, Laguerre and Legendre); Fourier series, Fourier and Laplace transforms; Elements of complex analysis: Laurent series-poles, residues and evaluation of integrals; Elementary ideas about tensors; Introductory group theory, SU(2), O(3); Elements of computational techniques: roots of functions, interpolation, extrapolation, integration by trapezoid and Simpson’s rule, solution of first order differential equations using Runge-Kutta method; Finite difference methods; Elementary probability theory, random variables, binomial, Poisson and normal distributions.

II. Classical Mechanics

Newton’s laws; Phase space dynamics, stability analysis; Central-force motion; Two-body collisions, scattering in laboratory and centre-of-mass frames; Rigid body dynamics, moment of inertia tensor, non-inertial frames and pseudoforces; Variational principle, Lagrangian and Hamiltonian formalisms and equations of motion; Poisson brackets and canonical transformations; Symmetry, invariance and conservation laws, cyclic coordinates; Periodic motion, small oscillations and normal modes; Special theory of relativity, Lorentz transformations, relativistic kinematics and mass–energy equivalence.

III. Electromagnetic Theory

Electrostatics: Gauss’ Law and its applications; Laplace and Poisson equations, boundary value problems; Magnetostatics: Biot-Savart law, Ampere's theorem, electromagnetic induction; Maxwell's equations in free space and linear isotropic media; boundary conditions on fields at interfaces; Scalar and vector potentials; Gauge invariance; Electromagnetic waves in free space, dielectrics, and conductors; Reflection and refraction, polarization, Fresnel’s Law, interference, coherence, and diffraction; Dispersion relations in plasma; Lorentz invariance of Maxwell’s equations; Transmission lines and wave guides; Dynamics of charged particles in static and uniform electromagnetic fields; Radiation from moving charges, dipoles and retarded potentials.

IV. Quantum Mechanics

Wave-particle duality; Wave functions in coordinate and momentum representations; Commutators and Heisenberg's uncertainty principle; Matrix representation; Dirac’s bra and ket notation; Schroedinger equation (time-dependent and time-independent); Eigenvalue problems such as particle-in-a-box, harmonic oscillator, etc.; Tunneling through a barrier; Motion in a central potential; Orbital angular momentum, Angular momentum algebra, spin; Addition of angular momenta; Hydrogen atom, spin-orbit coupling, fine structure; Time-independent perturbation theory and applications; Variational method; WKB approximation;Time dependent perturbation theory and Fermi's Golden Rule; Selection rules; Semi-classical theory of radiation; Elementary theory of scattering, phase shifts, partial waves, Born approximation; Identical particles, Pauli's exclusion principle, spin-statistics connection; Relativistic quantum mechanics: Klein Gordon and Dirac equations.

V. Thermodynamic and Statistical Physics

Laws of thermodynamics and their consequences; Thermodynamic potentials, Maxwell relations; Chemical potential, phase equilibria; Phase space, micro- and macrostates; Microcanonical, canonical and grand-canonical ensembles and partition functions; Free Energy and connection with thermodynamic quantities; First- and second-order phase transitions; Classical and quantum statistics, ideal Fermi and Bose gases; Principle of detailed balance; Blackbody radiation and Planck's distribution law; Bose-Einstein condensation; Random walk and Brownian motion; Introduction to nonequilibrium processes; Diffusion equation.

VI. Electronics

Semiconductor device physics, including diodes, junctions, transistors, field effect devices, homo and heterojunction devices, device structure, device characteristics, frequency dependence and applications; Optoelectronic devices, including solar cells, photodetectors, and LEDs; High-frequency devices, including generators and detectors; Operational amplifiers and their applications; Digital techniques and applications (registers, counters, comparators and similar circuits); A/D and D/A converters; Microprocessor and microcontroller basics.

VII. Experimental Techniques and data analysis

Data interpretation and analysis; Precision and accuracy, error analysis, propagation of errors, least squares fitting, linear and nonlinear curve fitting, chi-square test; Transducers (temperature, pressure/vacuum, magnetic field, vibration, optical, and particle detectors), measurement and control; Signal conditioning and recovery, impedance matching, amplification (Op-amp based, instrumentation amp, feedback), filtering and noise reduction, shielding and grounding; Fourier transforms; lock-in detector, box-car integrator, modulation techniques.Applications of the above experimental and analytical techniques to typical undergraduate and graduate level laboratory experiments.

VIII. Atomic & Molecular Physics

Quantum states of an electron in an atom; Electron spin; Stern-Gerlach experiment; Spectrum of Hydrogen, helium and alkali atoms; Relativistic corrections for energy levels of hydrogen; Hyperfine structure and isotopic shift; width of spectral lines; LS & JJ coupling; Zeeman, Paschen Back & Stark effect; X-ray spectroscopy; Electron spin resonance, Nuclear magnetic resonance, chemical shift; Rotational, vibrational, electronic, and Raman spectra of diatomic molecules; Frank – Condon principle and selection rules; Spontaneous and stimulated emission, Einstein A & B coefficients; Lasers, optical pumping, population inversion, rate equation; Modes of resonators and coherence length.

IX. Condensed Matter Physics

Bravais lattices; Reciprocal lattice, diffraction and the structure factor; Bonding of solids; Elastic properties, phonons, lattice specific heat; Free electron theory and electronic specific heat; Response and relaxation phenomena; Drude model of electrical and thermalconductivity; Hall effect and thermoelectric power; Diamagnetism, paramagnetism, and ferromagnetism; Electron motion in a periodic potential, band theory of metals, insulators and semiconductors; Superconductivity, type – I and type - II superconductors, Josephson junctions; Defects and dislocations; Ordered phases of matter, translational and orientational order, kinds of liquid crystalline order; Conducting polymers; Quasicrystals.

X. Nuclear and Particle Physics

Basic nuclear properties: size, shape, charge distribution, spin and parity; Binding energy, semi-empirical mass formula; Liquid drop model; Fission and fusion; Nature of the nuclear force, form of nucleon-nucleon potential; Charge-independence and charge-symmetry of nuclear forces; Isospin; Deuteron problem; Evidence of shell structure, single- particle shell model, its validity and limitations; Rotational spectra; Elementary ideas of alpha, beta and gamma decays and their selection rules; Nuclear reactions, reaction mechanisms, compound nuclei and direct reactions; Classification of fundamental forces; Elementary particles (quarks, baryons, mesons, leptons); Spin and parity assignments, isospin, strangeness; Gell-Mann-Nishijima formula; C, P, and T invariance and applications of symmetry arguments to particle reactions, parity non-conservation in weak interaction; Relativistic kinematics.