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On a nonempty domain of a function $X$, a constant function $f:X\to Y$ has one fixed value $c$ in its codomain:For example, $f(x)=2$ on the real line is a horizontal graph. Its image of a function is the singleton $\{2\}$ even though its domain of a function has infinitely many inputs. A constant function from the real line to the real numbers has derivative zero and every nonzero real period of a function; there is no smallest positive period. It supplies a reference level from which variation can be measured.

A constant map between vector spaces is a linear map only when its value is the zero vector, since a linear map must preserve zero. A nonzero constant map is still an affine map. In a Fourier series, the constant term records the period average, while the other terms describe oscillation around that level.

For a damped harmonic oscillator $\ddot{y}+\gamma \dot{y}+{\omega }_0^{2}y=0$, differentiation of the mechanical energy $E=m({\dot{y}}^2+{\omega }_0^2{y}^2)/2$ gives $\dot{E}=-m\gamma {\dot{y}}^2$. This identity is exact. In the weak-damping regime $\gamma \ll {\omega }_0$, the leading period average has equal kinetic energy and potential energy, so $\overline{{\dot{y}}^2}\simeq \overline{E}/m$ and $\dot{\overline{E}}\simeq -\gamma \overline{E}$. Thus the averaged energy has exponential decay, while the instantaneous energy has a small oscillatory correction. Its derivative vanishes at every turning point, ruling out an exact pointwise nonconstant exponential. The quality factor is $Q\simeq {\omega }_0/\gamma$ in this approximation.

In the prescribed along-beam scalar model $U=A(\xi )\cos \omega t$ with $A=U_0{e}^{-\lambda \xi }$, the first-order fluid displacement is $A\sin \omega t/\omega$. Its leading sampled-velocity correction is $-\lambda A^2\sin (2\omega t)/(2\omega )$, whose period average vanishes. This result concerns the supplied component model and does not determine an Eulerian mean streaming flow.

The characteristic equation of the damped harmonic oscillator is $r^2+\gamma r+{\omega }_0^2=0$. In the underdamped case its roots are $-\gamma /2\pm i{\omega }_d$, where ${\omega }_d=\sqrt{{\omega }_0^2-{\gamma }^2/4}$. Thus the exact real solution is $y=A{e}^{-\gamma t/2}\cos ({\omega }_{d}t+\alpha )$. For $\gamma \ll {\omega }_0$, ${\omega }_d={\omega }_0+O({\gamma }^2/{\omega }_0)$, giving the requested leading approximationFor mass $m$ and spring constant $m{\omega }_0^2$, the mechanical energy is $E=\frac{1}{2}m({\dot{y}}^2+{\omega }_0^2{y}^2)$. Differentiating and using the differential equation gives $\dot{E}=-m\gamma {\dot{y}}^2$. In particular, energy decay of a viscously damped harmonic oscillator is not an exact pointwise exponential decay: $\dot{E}=0$ at a turning point. To leading order in $\gamma /{\omega }_0$, the period average isThe quality factor is $Q=2\pi \overline{E}/(\text{energy lost in one cycle})$. Using the period $T\simeq 2\pi /{\omega }_0$ and the small fractional loss $\gamma T$ gives $\boxed{Q\simeq {\omega }_0/\gamma }$. Equivalently, the energy-decay definition $Q={\omega }_0\overline{E}/{\overline{P}}_{\mathrm{loss}}$ yields the same leading result. Replacing ${\omega }_d$ by ${\omega }_0$ in the oscillatory phase is not uniform over arbitrarily long times; its phase error is $O({\gamma }^{2}t/{\omega }_0)$.

For conservative forcing in averaged oscillator equations, $f=f(y)$, the average determining $R_T$ vanishes. If $F^{\prime }=f$, thenwhose period average is zero. Thus $R=R_0$ to this order, while a position-dependent force generally shifts the oscillation phase and frequency. The exact equation also conserves the energy $\frac{1}{2}{\dot{y}}^2+\frac{1}{2}{y}^2-\epsilon F(y)$, consistent with the absence of amplitude drift for bounded orbits.

For the cubic position force, $⟨f\cos \varphi ⟩=R^3⟨{\cos }^4\varphi ⟩=3R^3/8$. ThereforeThe leading oscillation is $R_0\cos [(1-3\epsilon R_0^2/8)t+{\theta }_0]$. For positive $\epsilon$ it is a softening cubic oscillator, obtained by weakly perturbing a harmonic oscillator. The weak-force condition is $\epsilon R_0^2\ll 1$; the constant $R$ is the leading harmonic amplitude, not a claim that the exact waveform remains sinusoidal.

Introduce the slow time $T=\epsilon t$ and treat $t,T$ independently in the method of multiple scales. WriteAt the next order,For fixed $R$ define the period average byHere $f$ inside the averages is evaluated at the leading position and velocity $R\cos \varphi ,-R\sin \varphi$. Orthogonality to both fundamental harmonics is the solvability condition in the method of multiple scales: it removes the resonant forcing that would otherwise generate a secular term. Since the squared sine and cosine averages are $1/2$, the amplitude-phase equations areEquivalently $\dot{R}=-\epsilon ⟨f\sin (t+\theta )⟩$ and $\dot{\theta }=-\epsilon ⟨f\cos (t+\theta )⟩/R$ at first order. These equations describe a bounded, weakly perturbed oscillation over $t=O({\epsilon }^{-1})$ while the amplitude remains in a range where the expansion is ordered. At zero amplitude the oscillation phase coordinate is singular; Cartesian harmonic coefficients or an equilibrium analysis should replace it.

Take the oscillation to be $z(t)=L\cos \omega t$; a different initial phase does not affect a period average. In the mass-density normalization of the quadrupole formula, the second mass moment tensor of the point mass has only $I_{33}=m{z}^2$ nonzero. Its trace-free mass quadrupole moment is thereforeThe constant part does not radiate. Since $d^3({\cos }^2\omega t)/d{t}^3=4{\omega }^3\sin 2\omega t$, the tensor contraction isUsing $⟨{\sin }^2(2\omega t)⟩=1/2$ in the quadrupole formula gives the averaged positive radiated powerThe mass quadrupole moment oscillates at twice the source's angular frequency, and the source energy loss has the opposite sign. The expression uses the leading slow-motion weak-field approximation, $\omega L\ll c$. With SI stress-energy tensor components, the mass-density integrand is $T_{00}/c^2$; the displayed $c^{-5}$ quadrupole formula uses that mass normalization. This is the point-mass contribution requested by the model: an apparatus maintaining an accelerating mass would also contribute its own mass quadrupole moment.

The period average extracts the reference level of an integrable function that repeats. For a real- or complex-valued periodic function $f$ with positive period of a function $T$, assume $f$ is Lebesgue integrable on every bounded interval. Its average isThe answer is independent of the starting point $a$. Splitting a shifted interval at the old endpoint and using $f(t+T)=f(t)$ moves its extra piece back into the missing piece. The interval length must be a period; an arbitrary window can give a different average.

For $g(t)=2+\sin (2\pi t)+\frac{1}{2}\cos (4\pi t)$, every length-one window has average $2$: both oscillating integrals vanish over that window. A half-unit window instead has average $2+(2/\pi )\cos (2\pi a)$, which depends on its start. Thus a short sample of an oscillation can bias an estimate of its reference level even when the full-cycle average is known exactly.

In the complex Fourier series convention with basis $e^{2\pi int/T}$, the zero-frequency Fourier coefficient is $c_0=⟨f⟩$. This assertion defines the coefficient even when an additional convergence theorem is needed to reconstruct $f$ from its series. Strang's §4.1 explicitly identifies that constant coefficient with the average.^[ref]

Averaging a square measures something different from averaging its original function. For the illustrated $g$, orthogonality of the trigonometric terms gives $⟨g^2⟩=4+1/2+1/8=37/8$, whereas $⟨g{⟩}^2=4$. Its root mean square is therefore $\sqrt{37/8}$. Similarly, the period average of ${\sin }^2$ is $1/2$. A zero mean need not imply a zero mean square.

A periodic function repeats exactly after a nonzero shift of its input. For $f:\mathbb{R}\to Y$, a positive period of a function $T$ satisfiesOne interval of length $T$ therefore determines the whole function. For another domain of a function $D$, a period must also preserve the domain, $D+T=D$; equality only where two domains happen to overlap is weaker. Periodicity is an exact translation of a function symmetry, rather than merely two similar-looking pieces of its graph.

The sine $\sin (2\pi x)$ has fundamental period one. The function ${\sin }^2(2\pi x)$ repeats after half a unit: the sine changes sign under that shift, but squaring removes the sign. The mixed function $g(x)=2+\sin (2\pi x)+\frac{1}{2}\cos (4\pi x)$ again has fundamental period one. All positive integer multiples of a period are periods, so specifying one period need not identify the shortest repetition.

If the input is time and $T_0$ is the fundamental period, the frequency is $1/T_0$ and the angular frequency is $2\pi /T_0$. Repetition lets a Fourier series describe a cycle using sines and cosines; the period average separates the reference level from the oscillating terms. Strang develops this representation and its Fourier coefficients in §4.1 of Computational Science and Engineering by Gilbert Strang.^[ref]

This connection grew out of a physical problem. In his 1822 book Théorie analytique de la chaleur by Joseph Fourier, Joseph Fourier used periodic functions to extend profiles beyond their initial interval while treating heat conduction. Chapter III, §§233–234 combines even functions and odd functions into sine–cosine expansions and rescales their period.^[ref] A modern Fourier series interpretation still requires the appropriate convergence conditions.

A constant function has every real shift as a period and no fundamental period. Without continuity, even a nonconstant periodic function can lack a smallest positive period; periodic function without a least positive period gives a concrete example. The period group of a function organizes all these possibilities.