Physics Background

Understanding the physics behind ultrasonic distance measurement.

Ultrasonic Sound

What is Ultrasonic?

Ultrasonic refers to sound waves with frequencies above the human hearing range.

• Human hearing: 20 Hz to 20 kHz
• Ultrasonic range: >20 kHz
• HC-SR04 frequency: 40 kHz

Why 40 kHz?

1. Above human hearing - No audible noise
2. Good directivity - Focused beam pattern
3. Reasonable range - Balance of range and accuracy
4. Low cost - Standard piezoelectric transducers available

Speed of Sound

Basic Formula

speed = distance / time

or

distance = speed × time

Speed of Sound in Air

At standard conditions (20°C, 1 atm):

v = 343 m/s
  = 0.343 m/ms
  = 0.0343 cm/µs
  = 1235 km/h
  = 767 mph

Temperature Dependence

Speed of sound varies with temperature:

v = 331.3 + (0.606 × T) m/s

Where T is temperature in Celsius.

Examples:

T = 0°C:   v = 331.3 m/s
T = 10°C:  v = 337.4 m/s
T = 20°C:  v = 343.0 m/s  ← Used by library
T = 30°C:  v = 349.1 m/s
T = 40°C:  v = 355.2 m/s

Effect on accuracy:

• At 0°C vs 20°C: ~3.4% slower
• At 40°C vs 20°C: ~3.6% faster

Distance Measurement Principle

Time-of-Flight

HC-SR04 uses Time-of-Flight (ToF) measurement:

1. Send ultrasonic pulse
2. Pulse travels to object
3. Pulse reflects back
4. Measure total time
5. Calculate distance

Round-Trip Distance

Total time = time to object + time back
           = 2 × (distance / speed)

Therefore:
distance = (total time × speed) / 2

Formula Derivation

Starting with:

distance = (time × speed) / 2

Using v = 343 m/s = 0.0343 cm/µs:

distance_cm = (time_µs × 0.0343) / 2
            = time_µs × 0.01715
            = time_µs / 58.31

Library uses 58.8235 (more precise):

distance_cm = time_µs / 58.8235

HC-SR04 Operation

Measurement Cycle

1. Trigger Pulse (10µs)
   ┌──┐
   │  │
───┘  └───────────

2. Ultrasonic Burst (8 pulses at 40kHz)
   ┌┐┌┐┌┐┌┐┌┐┌┐┌┐┌┐
   ││││││││││││││││
   └┘└┘└┘└┘└┘└┘└┘└┘
   |<-- 200µs -->|

3. Echo Wait
   Sound travels to object and back

4. Echo Pulse
   ┌────────────┐
   │            │
───┘            └───
   |<-- pulse -->|
   Duration = distance measurement

Timing Details

Trigger:

• Minimum pulse width: 10µs
• Typical: 10µs
• Initiates measurement

Ultrasonic Burst:

• 8 cycles of 40kHz
• Duration: 200µs (8 / 40000 Hz)
• Emitted automatically after trigger

Echo Pulse:

• Width proportional to distance
• Formula: width_µs = distance_cm × 58.8235

Beam Pattern

HC-SR04 has a cone-shaped detection pattern:

         Sensor
           ||
          /  \
         /    \
        /      \
       /  15°   \
      /          \
     /            \
    /      30°     \
   /                \
  /                  \
 
Effective angle: ~15°
Max angle: ~30°

Implications:

• Object must be within beam
• Small objects may be missed
• Angled surfaces may not reflect

Physical Limitations

Minimum Distance (~2 cm)

Causes:

1. Transducer recovery time - Takes time to switch from TX to RX
2. Blind zone - Residual vibration during echo wait
3. Close reflections - Pulse overlaps with echo

Maximum Distance (~4-7 m)

Limiting factors:

1. Signal attenuation - Sound weakens with distance
2. Air absorption - Higher frequency = more absorption
3. Ambient noise - Reflections, interference
4. Transducer sensitivity - Limited dynamic range

Accuracy (±3mm typical)

Factors:

1. Timing resolution - Arduino: 4µs → ~0.7mm
2. Temperature variation - ±5°C → ±1.5% error
3. Air pressure - Minor effect
4. Humidity - Minor effect at 40kHz

Environmental Factors

Temperature

Effect: Changes speed of sound

Error = (v_actual - v_assumed) / v_assumed × 100%

Example at 30°C:
Error = (349.1 - 343.0) / 343.0 × 100%
      = 1.78%

At 100cm: Error = ±1.78cm

Humidity

Effect: Slight increase in speed (~0.1-0.3%)

Negligible for most applications.

Air Pressure

Effect: Minimal at normal variations

Only significant at extreme altitudes.

Wind

Effect: Adds/subtracts from sound speed

v_effective = v_sound ± v_wind

At 10 m/s wind, 100cm distance:
Error ≈ ±2.9%

Surface Properties

Reflectivity

Different materials reflect sound differently:

Material	Reflectivity	Notes
Hard plastic	Excellent	Best
Metal	Excellent	Flat surfaces
Wood	Good	Dense wood better
Cardboard	Fair	Absorbs some sound
Fabric	Poor	Absorbs sound
Foam	Very poor	Highly absorbent

Angle of Incidence

Sound reflects like light:

Angle of incidence = Angle of reflection

Perpendicular (90°): Best reflection
      ||
      \/
    ──────
    
Angled (45°): Reflects away
      ||
     /  \
    /    \
   ──────

Best practice: Mount sensor perpendicular to target (±15°)

Calculation Examples

Example 1: Echo Time to Distance

Given: Echo pulse width = 1470 µs

distance_cm = 1470 / 58.8235
            = 25.0 cm

Example 2: Distance to Echo Time

Given: Want to measure 100 cm

time_µs = 100 × 58.8235
        = 5882.35 µs
        ≈ 5882 µs

Example 3: Temperature Compensation

Measured at 20°C: 100.0 cm Actual temp: 30°C

v_20C = 343.0 m/s
v_30C = 331.3 + (0.606 × 30) = 349.1 m/s

Correction = v_30C / v_20C
           = 349.1 / 343.0
           = 1.0178

Corrected distance = 100.0 × 1.0178
                   = 101.78 cm

Example 4: Maximum Range Calculation

Timeout: 20000 µs

max_range_cm = 20000 / 58.8235
             = 340.0 cm
             = 3.4 m

Example 5: Timeout for Desired Range

Want: 2 meter range

timeout_µs = 200 cm × 58.8235
           = 11764.7 µs
           ≈ 11765 µs

Ultrasonic vs Other Technologies

Comparison

Technology	Range	Accuracy	Cost	Power
Ultrasonic	2-400cm	±3mm	Low	Low
IR Sharp	10-80cm	±5mm	Low	Low
Laser ToF	5-400cm	±1mm	High	Medium
Radar	10-5000cm	±1cm	High	Medium

When to Use Ultrasonic

Advantages:

• Low cost
• Simple interface
• Works in dust/fog
• Wide beam (finds objects easily)

Disadvantages:

• Temperature sensitive
• Affected by soft surfaces
• Slow update rate
• Limited range

Advanced Topics

Doppler Effect

Moving objects shift frequency:

f_observed = f_source × (v_sound ± v_observer) / (v_sound ± v_source)

HC-SR04 doesn't measure frequency, so Doppler doesn't affect distance reading.

Multi-path Reflections

Sound can bounce multiple times:

Direct: Sensor → Object → Sensor
Multi-path: Sensor → Wall → Object → Wall → Sensor

Can cause incorrect readings.

Mitigation: Use shortest measured distance

Interference

40kHz is common frequency:

• Other ultrasonic sensors
• Some electronic ballasts
• Animal calls (bats)

Mitigation: Time multiplex multiple sensors

Mathematical Formulas

Core Formulas

Distance from time:

d = (t × v) / 2

Time from distance:

t = (2 × d) / v

CM conversion:

d_cm = t_µs / 58.8235

Temperature-corrected speed:

v = 331.3 + (0.606 × T)

Unit Conversions

CM to other units:

meters = cm × 0.01
mm = cm × 10.0
inches = cm × 0.393701
yards = cm × 0.0109361
miles = cm × 0.000006213712

Practical Considerations

Measurement Rate

Minimum time between readings:

t_min = 2 × (d_max / v) + overhead
      = 2 × (4m / 343 m/s) + 20ms
      ≈ 43ms

Practical minimum: 60ms
Maximum rate: ~16 Hz

Power Consumption

HC-SR04 typical:

• Idle: <2 mA
• During ping: ~15 mA for ~200µs
• Average (1 Hz): ~2-3 mA
• Average (10 Hz): ~3-5 mA

Resolution

Theoretical resolution:

Arduino timing resolution: 4µs (16MHz)
Distance resolution: 4µs / 58.8235 ≈ 0.068 cm

Practical resolution: ~3mm (0.3cm)

See Also

• Unit Conversions - Detailed conversion math
• Constants - Physical constants used
• Performance - Performance characteristics
• Architecture - Implementation details