 Solution

In pulse-echo ultrasound, a transducer emits a pulse and records returning echoes. For a straight round-trip path in a medium of assumed sound speed $c$, an echo delay $t$ gives depth $z=ct/2$. Reflections depend on acoustic-impedance changes, while attenuation, scattering and propagation geometry affect the received amplitude.

An A-mode ultrasound scan displays echo amplitude against inferred depth along one beam. A B-mode ultrasound scan maps received amplitude to brightness and assembles many beam directions or translated beam lines into a spatial image. An M-mode ultrasound scan repeats one beam line and displays its depth profile versus time, making motion along that line visible. Thus the display axes and acquisition pattern differ even when the pulse-echo principle is shared.

For a pulse lasting ${\tau }_p$, a simple axial-separation estimate is $c{\tau }_p/2$. Lateral resolution depends on aperture, wavelength, focusing and depth. Shorter pulses and broader bandwidth can improve axial resolution; an assumed sound speed or a multiple-reflection path can instead misplace an echo. Brightness is not a direct universal map of composition.

Doppler ultrasound uses frequency or pulse-to-pulse phase changes from moving scatterers. In a simple backscatter geometry, $\Delta f\simeq 2f_{0}v\cos \theta /c$. Inferring speed requires the beam-to-motion angle; pulsed sampling can alias frequencies, and an unknown angle leaves a velocity-projection ambiguity. This describes generic acoustic acquisition rather than a clinical diagnosis.