I'm currently studying the textbook Fundamentals of Acoustics (2000) by Kinsler et al. Chapter 1.2 The Simple Oscillator says the following:
$$\dfrac{d^2 x}{dt^2} + \omega_0^2 x = 0 \tag{1.2.5}$$
This is an important linear differential equation whose general solution is well known and may be obtained by several methods.
One method is to assume a trial solution of the form
$$x = A_1 \cos(\gamma t) \tag{1.2.6}$$
Differentiation and substitution into (1.2.5) shows that this is a solution if $\gamma = \omega_0$. It may similarly be shown that
$$x = A_2 \sin(\omega_0 t) \tag{1.2.7}$$
is also a solution. The complete general solution is the sum of these two,
$$x = A_1 \cos(\omega_0 t) + A_2 \sin(\omega_0 t) \tag{1.2.8}$$
where $A_1$ and $A_2$ are arbitrary constants and the parameter $\omega_0$ is the natural angular frequency in radians per second (rad/s).
Chapter 1.3 Initial Conditions says the following:
If at time $t = 0$ the mass has an initial displacement $x_0$ and an initial speed $u_0$, then the arbitrary constants $A_1$ and $A_2$ are fixed by these initial conditions and the subsequent motion of the mass is completely determined. Direct substitution into (1.2.8) of $x = x_0$ at $t = 0$ will show that $A_1$ equals the initial displacement $x_0$. Differentiation of (1.2.8) and substitution of the initial speed at $t = 0$ gives $u_0 = \omega_0 A_2$, and (1.2.8) becomes
$$x = x_0 \cos(\omega_0 t) + (u_0/\omega_0) \sin(\omega_0 t) \tag{1.3.1}$$
Another form of (1.2.8) may be obtained by letting $A_1 = A\cos(\phi)$ and $A_2 = -A\sin(\phi)$, where $A$ and $\phi$ are two new arbitrary constants. Substitution and simplification then gives
$$x = A\cos(\omega_0 t + \phi) \tag{1.3.2}$$
where $A$ is the amplitude of the motion and $\phi$ is the initial phase angle of the motion. The values of $A$ and $\phi$ are determined by the initial conditions and are
$$A = [x_0^2 + (u_0/\omega_0)^2]^{1/2} \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \phi = \tan^{-1}(-u_0/\omega_0 x_0) \tag{1.3.3}$$
Successive differentiation of (1.3.2) shows that the speed of the mass is
$$u = -U \sin(\omega_0 t + \phi) \tag{1.3.4}$$
where $U = \omega_0 A$ is the speed amplitude, and the acceleration of the mass is
$$a = – \omega_0 U \cos(\omega_0 t + \phi) \tag{1.3.5}$$
In these forms it is seen that the displacement lags $90^\circ$ ($\pi/2$ rad) behind the speed and that the acceleration is $180^\circ$ ($\pi$ rad) out of phase with the displacement, as shown in Fig. 1.3.1.
(Arrows in figure 1.3.1 are mine.)
We can see from figure 1.3.1 that the displacement is out of phase with the acceleration by $\pi$ radians (green arrow), as stated, but it seems to me that, according to figure 1.3.1, displacement is actually $3\pi/2$ radians out of phase with speed (blue arrow), rather than the stated $\pi/2$ radians (red arrow). Is this an error, or am I misunderstanding this?
Best Answer
Note that in the diagram below the velocity leads the displacement by $\dfrac \pi 2$ which is the same as the velocity lagging the displacement by $\dfrac{3\pi}{2}$.
So when mentioning phase it is important to state which two quantities are being compared, eg $A$ and $B$, and then whether there is a lead or lag between them, eg $A$ leads/lags $B$.
$A$ leading $B$ by $\phi$ is the same as $A$ lagging $B$ by $2\pi -\phi$.