Fluids & Waves

Fluids & Waves

Part 1 of 4 — Fluid Statics: Density, Pressure & Buoyancy

Density and Specific Gravity

$\rho = \frac{m}{V} \qquad SG = \frac{\rho_{substance}}{\rho_{water}}$

Water's density is the anchor: $1000\;\text{kg/m}^3 = 1\;\text{g/cm}^3$. Specific gravity is dimensionless — $SG = 0.8$ means 80% as dense as water. Memorize mercury's $SG = 13.6$.

Pressure

$P = \frac{F}{A}$

Pressure is a scalar (no direction), in pascals ($1\;\text{Pa} = 1\;\text{N/m}^2$); $1\;\text{atm} \approx 10^5\;\text{Pa} = 760\;\text{mmHg}$. Beneath a fluid surface:

$P = P_0 + \rho g h$

The gauge pressure $\rho g h$ is the pressure above atmospheric; absolute pressure adds $P_0$. Depth pressure depends ONLY on depth, fluid density, and $g$ — never on the container's shape or total volume. A skinny standpipe and a lake apply identical pressure at the same depth.

Pascal's Principle: The Hydraulic Lever

Pressure applied to an enclosed fluid transmits undiminished everywhere. In a hydraulic lift, equal pressures at both pistons mean

$\frac{F_1}{A_1} = \frac{F_2}{A_2}$

A small force on a small piston yields a large force on a large piston — but the small piston must travel proportionally farther ($A_1d_1 = A_2d_2$), so the work is never multiplied. It is the fluid version of a lever.

Archimedes' Principle

$F_B = \rho_{fluid} \, V_{submerged} \, g$

The buoyant force equals the weight of displaced fluid — fluid density, not object density, and only the submerged volume counts. Two consequences:

• A FLOATING object displaces its own weight of fluid; the fraction submerged equals its specific gravity ($SG = 0.9$ iceberg floats 90% underwater)
• A fully SUBMERGED object displaces its own volume; it sinks if $\rho_{object} > \rho_{fluid}$, and the buoyant force does not change with depth (incompressible fluid)

Worked Example — Hydraulic Lift

A mechanic pushes down with $100\;\text{N}$ on a piston of area $0.01\;\text{m}^2$. The output piston has area $0.1\;\text{m}^2$.

Step 1 — Pressure created.

$P = \frac{100}{0.01} = 10^4\;\text{Pa}$

Step 2 — Force on the large piston.

$F_2 = PA_2 = (10^4)(0.1) = 1000\;\text{N}$

A tenfold force multiplication from the tenfold area ratio. To raise the load $0.05\;\text{m}$, the mechanic's piston must sweep $0.5\;\text{m}$ — work in equals work out ($50\;\text{J}$ both ways).

Worked Example — Floating and Reading Specific Gravity

A block of wood floats in water with 60% of its volume submerged.

Floating means $F_B = W$:

$\rho_{water}(0.6V)g = \rho_{wood}Vg \Rightarrow \rho_{wood} = 0.6\,\rho_{water} = 600\;\text{kg/m}^3$

The submerged fraction IS the specific gravity. Moved to oil ($SG = 0.8$), the same block must displace more oil to match its weight: fraction $= 0.6/0.8 = 0.75$, so it rides lower. In a fluid with $SG < 0.6$ it would sink entirely.

Gauge check: a diver at $20\;\text{m}$ in fresh water ($g = 10\;\text{m/s}^2$) feels gauge pressure $\rho g h = (1000)(10)(20) = 2 \times 10^5\;\text{Pa}$ — about 2 atm above the surface, 3 atm absolute.

Fluid Statics 🎯

Key Takeaways — Part 1

• $SG = \rho/\rho_{water}$; water is $1000\;\text{kg/m}^3$; submerged fraction of a floater = its SG
• Hydrostatic pressure $P = P_0 + \rho gh$ depends on depth and density only — never container shape
• Gauge pressure excludes atmosphere; absolute includes it; roughly +1 atm per 10 m of water
• Pascal: $F_1/A_1 = F_2/A_2$ — hydraulics multiply force, never work
• Buoyant force = weight of DISPLACED fluid ($\rho_{fluid}V_{sub}g$), independent of depth
• Sink or float: compare object density to fluid density

Fluids & Waves

Part 2 of 4 — Fluid Dynamics: Continuity, Bernoulli & Viscosity

Continuity: Mass In = Mass Out

For an incompressible fluid in a pipe, the volumetric flow rate $Q$ is the same everywhere:

$Q = A_1 v_1 = A_2 v_2$

Narrow the pipe and the fluid MUST speed up. Since area scales with radius squared, halving the radius quarters the area and quadruples the speed.

Bernoulli: Energy Conservation for Ideal Fluids

$P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}$

Valid for ideal flow: incompressible, nonviscous, laminar. At constant height, the trade is stark: where the fluid moves fast, its pressure is low. This feels backward — students picture fast fluid "blasting" harder — but the pressure term is what remains after the kinetic term takes its share.

Discriminate the two tools: continuity converts areas into speeds; Bernoulli converts speeds into pressures. A classic MCAT chain uses both: area down, so speed up (continuity), so pressure down (Bernoulli).

A special case worth knowing: Torricelli's theorem. Fluid leaking from a hole a depth $h$ below the open surface exits at $v = \sqrt{2gh}$ — the free-fall speed, because Bernoulli here is just energy conservation per unit volume.

Viscosity: When Fluids Are Real

Real fluids dissipate energy through internal friction (viscosity $\eta$), so pressure drops along a pipe even at constant area. Poiseuille's law for laminar viscous flow:

$Q = \frac{\pi \, \Delta P \, r^4}{8 \eta L}$

The punchline is the fourth power of the radius: doubling $r$ multiplies flow 16-fold; a 19% narrowing already halves it. This is why arteriolar radius is the body's dominant flow control and why mild vessel narrowing is hemodynamically expensive.

At high speeds flow turns turbulent — chaotic, energy-hungry, and audible (heart murmurs, bruits). High velocity, large diameter, high density, and low viscosity all push toward turbulence.

Worked Example — Chaining Continuity into Bernoulli

Water flows at $1\;\text{m/s}$ through a horizontal pipe of cross-section $10\;\text{cm}^2$ that narrows to $2\;\text{cm}^2$.

Step 1 — Continuity for the new speed.

$v_2 = v_1\frac{A_1}{A_2} = (1)\frac{10}{2} = 5\;\text{m/s}$

Step 2 — Bernoulli for the pressure change (horizontal, so the $\rho gh$ terms cancel):

$P_1 - P_2 = \tfrac{1}{2}\rho(v_2^2 - v_1^2) = \tfrac{1}{2}(1000)(25 - 1) = 12000\;\text{Pa}$

The narrow, fast section is LOWER in pressure by $12\;\text{kPa}$. Order matters: continuity first (geometry to speed), Bernoulli second (speed to pressure).

Worked Example — Torricelli's Tank

A drum of water has a small hole $5\;\text{m}$ below the open surface ($g = 10\;\text{m/s}^2$):

$v = \sqrt{2gh} = \sqrt{2(10)(5)} = 10\;\text{m/s}$

Same speed a dropped stone would have after falling $5\;\text{m}$ — the escaping parcel cashes in exactly its column's potential energy.

Viscous contrast: none of the above predicts a pressure drop along a UNIFORM pipe, yet real IV lines and blood vessels lose pressure steadily along their length. That loss is viscosity's signature, governed by Poiseuille — ideal-fluid tools cannot see it.

Fluid Dynamics 🎯

Key Takeaways — Part 2

• Continuity: $A_1v_1 = A_2v_2$; area shrinks as $r^2$, so half the radius means four times the speed
• Bernoulli (ideal flow only): fast means low pressure at the same height
• Standard chain: geometry to speed (continuity), then speed to pressure (Bernoulli)
• Torricelli: efflux speed $\sqrt{2gh}$, the free-fall speed from the surface
• Poiseuille for real, viscous flow: $Q \propto \Delta P \, r^4/(\eta L)$ — the $r^4$ is the tested fact
• Turbulence arrives at high speed and is noisy (murmurs); ideal-fluid equations stop applying

Fluids & Waves

Part 3 of 4 — Waves & Sound: Standing Waves, Decibels & Doppler

The Wave Equation and Who Controls What

$v = f\lambda$

The medium sets the speed (string tension and mass density; air temperature for sound). The source sets the frequency. Wavelength is the negotiated result. So when a wave crosses into a new medium, frequency stays fixed while speed and wavelength change together — the single most-tested wave fact.

Sound is longitudinal (compressions along the travel direction, ~$343\;\text{m/s}$ in air, faster in liquids and solids); light and waves on strings are transverse.

Superposition and Standing Waves

Overlapping waves add. In phase: constructive (bigger). Out of phase: destructive (cancellation). Two identical waves traveling opposite directions build a standing wave with fixed nodes (no motion) and antinodes (maximum motion).

The closed pipe hides two traps: its fundamental is an octave LOWER than an open pipe of the same length ($4L$ vs $2L$ wavelength), and even harmonics simply do not exist for it.

Intensity vs. Sound Level (Do Not Conflate)

Intensity $I$ (in $\text{W/m}^2$) is physical power per area, falling off as $1/r^2$ from a point source. Sound LEVEL $\beta$ is the logarithmic decibel scale:

$\beta = 10\log\frac{I}{I_0}, \qquad I_0 = 10^{-12}\;\text{W/m}^2$

Every $+10\;\text{dB}$ means intensity $\times 10$. So $+20\;\text{dB}$ is $\times 100$, $+30\;\text{dB}$ is $\times 1000$; doubling intensity adds only about $3\;\text{dB}$.

Doppler: Relative Approach Raises Pitch

$f_{obs} = f_{src}\,\frac{v \pm v_{obs}}{v \mp v_{src}}$

Sign ritual: choose each sign so that approach raises $f_{obs}$ and recession lowers it — top sign for approach in both numerator and denominator. A moving SOURCE physically compresses the wavelength ahead of it; a moving OBSERVER meets wavefronts more often but leaves $\lambda$ untouched.

Worked Example — A Closed Organ Pipe

A pipe $0.85\;\text{m}$ long is closed at one end ($v_{sound} = 340\;\text{m/s}$).

Step 1 — Fundamental. A closed pipe fits a quarter wavelength: $\lambda_1 = 4L = 3.4\;\text{m}$.

$f_1 = \frac{v}{4L} = \frac{340}{3.4} = 100\;\text{Hz}$

Step 2 — Next harmonic. Closed pipes skip even harmonics, so the next is the THIRD: $f_3 = 3f_1 = 300\;\text{Hz}$. (An open pipe of the same length would give $f_1 = 200\;\text{Hz}$ with the full series $200, 400, 600, ...$)

Worked Example — Decibel Arithmetic Without a Calculator

A monitor alarm measures $40\;\text{dB}$ at a nurse's station and $70\;\text{dB}$ beside the patient.

$\Delta\beta = 30\;\text{dB} \Rightarrow \frac{I_2}{I_1} = 10^{30/10} = 10^3 = 1000$

Thirty decibels is three factors of ten. Never treat dB differences as ratios ($70/40$) — the scale is logarithmic.

Worked Example — Doppler Direction Check

An ambulance siren at $600\;\text{Hz}$ approaches a stationary pedestrian at $34\;\text{m/s}$ ($v = 340\;\text{m/s}$):

$f_{obs} = 600\,\frac{340}{340 - 34} = 600\,\frac{340}{306} \approx 667\;\text{Hz}$

Approach, so the answer must exceed $600$ — the minus sign in the denominator is chosen to make that happen. After it passes, the denominator flips to $374$ and the pitch drops to about $545\;\text{Hz}$: the classic drive-by pitch fall.

Waves & Sound 🎯

Key Takeaways — Part 3

• $v = f\lambda$: medium sets speed, source sets frequency; crossing media changes $v$ and $\lambda$, never $f$
• Standing waves: strings and open pipes give all harmonics ($nv/2L$); closed pipes give odd only ($nv/4L$)
• A closed pipe's fundamental is an octave below an equal-length open pipe's
• Decibels are logarithmic: $+10\;\text{dB} = \times 10$ intensity; doubling intensity adds about $3\;\text{dB}$
• Intensity falls as $1/r^2$ from a point source
• Doppler: approach raises pitch, recession lowers it; pick the formula's signs to enforce that

Fluids & Waves

Part 4 of 4 — MCAT Integration: Blood Flow, Ultrasound & Passage Strategy

The Circulation as a Fluids Problem

The MCAT rarely asks about pipes; it asks about vessels. Map the physics onto the physiology:

• Continuity with TOTAL area: each capillary is microscopic, but there are billions — their summed cross-section (~$1000\times$ the aorta's) is what enters $Q = A_{total}v$. Blood therefore CRAWLS through capillaries, buying time for gas and nutrient exchange. Answering "blood speeds up in narrow capillaries" applies single-vessel intuition to a branching bed — the signature trap.
• Poiseuille for resistance: flow scales with $r^4$, so arteriolar constriction is the body's volume knob, and a modestly stenosed coronary artery loses most of its flow reserve.
• Bernoulli in pathology: blood racing through a stenosis is a low-pressure region; the vessel wall can flutter or transiently collapse, and downstream of the jet, turbulence makes the murmur you hear with a stethoscope.
• Hydrostatics in posture: standing adds $\rho gh$ to blood pressure below the heart — ankle venous pressure far exceeds brachial pressure, which is why measurement is standardized at heart level.

Ultrasound: Waves Doing Medicine

Ultrasound imaging reflects megahertz sound off tissue interfaces; Doppler ultrasound reads blood velocity from the frequency shift of echoes off moving red cells — approach shifts the echo up, recession shifts it down, and the shift size gives the speed. A stenotic jet announces itself as a localized velocity (and Doppler-shift) spike.

Passage Strategy: Pick the Regime First

1. Is the fluid treated as ideal (continuity, Bernoulli, Torricelli) or viscous (Poiseuille, pressure loss along a uniform tube)? The passage's assumptions decide.
2. Single tube or branching network? Networks demand total cross-sectional area.
3. For sound: is the question about intensity (watts, $1/r^2$) or level (dB, logarithmic)? Convert deliberately, never by proportion.
4. Doppler: identify WHO moves, then force the sign so approach raises frequency.

Worked Example — Why Capillary Blood Crawls

The aorta (cross-section $\approx 3\;\text{cm}^2$) carries blood at about $30\;\text{cm/s}$. The capillary bed's total cross-section is roughly $900\;\text{cm}^2$.

Step 1 — Flow rate is fixed by the heart.

$Q = Av = (3)(30) = 90\;\text{cm}^3/\text{s}$ (about 5.4 L/min — cardiac output, a sanity check)

Step 2 — Continuity across the whole bed.

$v_{cap} = \frac{Q}{A_{total}} = \frac{90}{900} = 0.1\;\text{cm/s}$

A 300-fold slowdown, in the one place slow flow is physiologically useful. Any single capillary IS narrower than the aorta, but continuity binds $Q$ to the summed area of the parallel paths.

Worked Example — Grading a Stenosis with Doppler

A carotid ultrasound shows peak systolic velocity jumping from $60\;\text{cm/s}$ upstream to $240\;\text{cm/s}$ within a stenosis.

Step 1 — Continuity backward. A $4\times$ speed-up at (approximately) fixed flow means the lumen area is down to one-quarter: $A \propto 1/v$. Radius is therefore halved ($A \propto r^2$) — about a 50% diameter stenosis.

Step 2 — Bernoulli forward. The kinetic term $\tfrac{1}{2}\rho v^2$ grows $16\times$ in the jet, so intravascular pressure dips there; post-stenotic turbulence produces the audible bruit.

Step 3 — Poiseuille's warning. With radius halved, the segment's flow capacity at a given pressure difference is down to $(1/2)^4 = 1/16$ — the physiologic cost is far steeper than "half blocked" sounds.

MCAT Integration: Fluids & Waves 🎯

Key Takeaways — Part 4

• Branching beds use TOTAL cross-sectional area in continuity: capillary blood crawls because the summed area is enormous
• One narrowed vessel: faster and lower-pressure inside the stenosis (continuity + Bernoulli); turbulence downstream is the murmur
• Poiseuille's $r^4$: halving radius cuts flow to $1/16$ — constriction is the circulation's volume knob
• Standing posture adds $\rho gh$ to pressures below the heart; measure BP at heart level
• Doppler ultrasound: approach shifts echoes up in frequency; the shift encodes blood speed
• Strategy: fix the regime first (ideal vs. viscous, single tube vs. network, intensity vs. dB), then compute