The time varied gain (TVG) of a sonar is intended to remove the range dependence of echo strength. The conventional "40 log R" and "20 log R" TVG functions, which apply to single and distributed targets respectively, provide exact compensation only at infinite range 9 At short range, the conventional functions are inexact due to bandwidth related delays and the change in receiver gain over a pulse length. The theory of echo formation is used to derive exact gain functions which make the echo energy integral independent of the target range. In the case of randomly distributed targets, the linear form of the exact function is shown to be q5 (t) = et exp (ctct/2)~/{(1 -TI/t) z -( T2/t)2}, for sound speed c and absorption coefficient t~. Tt and T2 are constants for a given sonar and target. The ct exp (act~2) term is equivalent to "20 log R + 2 aR". The single target function is similarly the conventional function multiplied by a polynomial expression in 1/t. Analytic functions are derived for systems with simple transfer functions. As the pulse length bandwidth product increases, the exact function tends to that of the wideband ideal system for which T~ = T]2 and T2 = T/~/(12), T being the transmitter pulse length. Exact TVG functions are derived numerically for two echo sounders used in fishery research and are compared Wit.h the measured gain variation. The TVG function realized in sonars may depart considerably from the exact form. Delaying the start of the TVG ramp may reduce the error. The delay required for exact compensation depends upon the target range and is at least half the pulse length.