HARMONIC ANALYSIS, The. °The Harmonic Analysis') is the name first given by Thomson and Tait in their 'Natural Philos ophy) to a method extensively and fruitfully employed in investigations in many branches of mathematical physics, and first used by Daniel Bernouilli and Euler in the middle of the 18th century in studying the musical vibra tions of a stretched elastic string.
From the physical side it is described by J. Clerk Maxwell as °a method by which the solu tion of an actual problem may be obtained as the sum or resultant of a number of terms, each of which is a solution of a particular case of the problem.° The method is applicable to physical problems the actual complicated state under investigation can be regarded as due to the superposition of a number of simpler states that can coexist without interfering with one another.
• For example, in dealing with the small oscil lations of a musical string it is known that the string is capable of sounding a variety of so called pure notes, known as the fundamental note, the first harmonic or octave of the funda mental note, and the higher harmonics of the fundamental note, and that the forms of vibra tion giving these various notes may coexist, so that the string may be sounding at once its fundamental note and its various harmonics and thus be giving a note quite distinguishable from its pure fundamental note though of the same pitch. If we are dealing with the problem of the motion of a string sounding such a com plicated note, the harmonic analysis enables us to obtain and to express its solution as a sum of the terms expressing the motions which separately would give the separate pure notes actually present.
From the mathematical side the problems to which the harmonic analysis is applicable are those in which it is necessary to find a solution of a homogeneous linear differential equation which shall satisfy a set -of given initial or boundary conditions sufficiently numerous to make the problem determinate. It is well known that if a solution of such a differential equation has been obtained, it may be multiplied by any constant and will still be a solution; and that if several solutions have been obtained, their sum will be a solution. In using the harmonic
analysis we attempt by a skilful use of these two principles to so combine simple particular solutions of the differential equation involved in the problem as to form a solution of the equation which satisfies all the given conditions. This usually makes it necessary to analyze some one of the given conditions into a sum or series of simpler so-called harmonic terms, or in other words to develop some function of one of the independent variables, or of a set of the inde pendent variables into a series whose terms are of specified form.
For instance, suppose a harp-string of length 1 initially distorted into a curve whose equa tion referred to the position of equilibrium of the string as the X-axis and to one end of the string as origin is y=f(x), and then released, and that it is required to solve the problem of the subsequent motion of the string, the initial displacement being small.
Here we have to solve the differential equa tion 8'y , a'y a — (I)subject to the conditions y;) when .2.-3; ay when x=l; 0 when t = 0; y f(x) whet at r=o. It is known and is easily verified that y = sin fix cos 4,3t is a particular solution of (I,' if is any constant. If we take ,9 , when X m is any whole number, y = sin Mr — cos InTra 1 is a solution of (I) which satisfies our first three conditions; and so is . rat 27rx 2rat Y= al — cos — a2 sin — cos — I 1 1 3rx . . , (1)+ al sin — nog 1 where at, a2, as, . . . are any constants When t==0 (1) reduces to 7rx 21tx, 3/rx , y -r sin -r a, ULU -r ... (2) 1 and if we can choose at, a,, etc., so that the series in (2) is equal to f(x) for all values of x between 0 and 1, (1) becomes our required solution. This calls for the development of f(x) into a Trigonometric Series of somewhat peculiar form known as a Fourier's Series, and when that is accomplished our solution is com plete.