Study sheet: Mechanical Waves

📋 Course Outline

1. Wave Propagation and Wavelength
2. Wavefronts and Wave Types
3. Source and Point Equations
4. Phase and Spatial Profiles
5. Phase Difference and Stroboscopy

📖 1. Wave Propagation and Wavelength

🔑 Key Concepts & Definitions

• Progressive wave : a phenomenon resulting from the propagation of a series of identical disturbances emitted regularly by a source
• Wavelength : the distance traveled by a wave during one temporal period $T$, and it depends on the propagating medium

★ Must-know

📐 Formula — The wavelength satisfies $\lambda = T V = \frac{V}{N}$, where $V$ is the wave speed and $N$ is the frequency.

Further detail

• Two consecutive water ridges of the same nature, both crests or both troughs, are separated by one wavelength $\lambda$.

💡 Memory Hook

Repeated source disturbances → progressive wave propagation

📖 2. Wavefronts and Wave Types

🔑 Key Concepts & Definitions

• Wavefront : the most advanced disturbance created by the source

★ Must-know

📌 The wavefront is a crest when the source motion begins upward, with source phase $\varphi_S=0$, and a trough when the source motion begins downward, with source phase $\varphi_S=\pi$.

📌 A transverse wave has propagation perpendicular to the deformation, as for a rope or the water surface, whereas a longitudinal wave has propagation parallel to the deformation, as for a spring or sound.

Further detail

• At a given instant, the point M1 can move in the negative direction of elongations while point M2 moves in the positive direction of elongations.

💡 Memory Hook

Transverse: perpendicular; longitudinal: parallel

📖 3. Source and Point Equations

★ Must-know

📐 Formula — The source elongation is described by $y_S(t)=a\sin(\omega t+\varphi_S)=a\sin(2\pi Nt+\varphi_S)$, with $\varphi_S=0$ or $\pi$.

📐 Formula — For a point M at abscissa $x$, the propagation delay is $\theta=x/V$ and its elongation is $y_M(t,x)=a\sin(\omega t-2\pi x/\lambda+\varphi_S)$.

📐 Formula — The angular frequency satisfies $\omega=2\pi N=2\pi/T$, and the phase of point M is $\varphi_M=-2\pi x/\lambda+\varphi_S$.

Further detail

• The propagation principle gives the point motion through the delayed source motion: $y_M(t)=y_S(t-\theta)$ and equivalently $y_S(t)=y_M(t+\theta)$.

💡 Memory Hook

Source motion → delayed point motion → propagated equation

📖 4. Phase and Spatial Profiles

★ Must-know

• For a fixed position, the time profile is sinusoidal after the disturbance arrives and is zero before arrival: $y_M(t)=a\sin(\omega t-2\pi x_0/\lambda+\varphi_S)$ for $t>\theta$ and $y_M(t)=0$ for $0<t<\theta$.

📐 Formula — The wavefront position at time $t_0$ is $x_f=Vt_0=t_0N\lambda$.

Further detail

📌 If the wavefront position $x_f$ is less than the rope length $L$, only part of the rope is affected; if $x_f>L$, the entire rope is affected.

• For a fixed time, the spatial profile of the rope is sinusoidal for positions before the wavefront and zero for positions beyond the wavefront.

💡 Memory Hook

Time profile varies with t; spatial profile varies with x

📖 5. Phase Difference and Stroboscopy

★ Must-know

📐 Formula — The phase difference from the source is $\Delta\varphi=\varphi_S-\varphi_M=2\pi x/\lambda$.

📌 A point is in phase with the source when $\Delta\varphi=2K\pi$, corresponding to $x=K\lambda$.

📌 In stroboscopy, if the stroboscopic period is $T_e=KT$, the apparent motion is stationary and the stroboscopic frequency is $N_e=N/K$.

Further detail

📌 A point is in phase quadrature advance when $\Delta\varphi=2K\pi-\pi/2$ and in phase quadrature delay when $\Delta\varphi=2K\pi+\pi/2$.

📌 If $T_e>KT$, then $N_e<N/K$ and the apparent motion is slowed in the real direction; if $T_e<KT$, then $N_e>N/K$ and it is slowed in the opposite direction.

💡 Memory Hook

In phase, opposite phase, and quadrature correspond to distinct separations

📊 Synthesis Tables

Phase Relationships

Relationship	Phase difference	Distance
In phase	$2K\pi$	$K\lambda$
Opposition of phase	$(2K+1)\pi$	$(2K+1)\lambda/2$
Quadrature advance	$2K\pi-\pi/2$	$(4K-1)\lambda/4$
Quadrature delay	$2K\pi+\pi/2$	$(4K+1)\lambda/4$