What is Wave Equation Analysis?

Wave Equation Analysis is a numerical analysis that represents the whole pile driving system as a chain of masses and springs, from the ram, hammer cushion, helmet and pile body to the surrounding soil. The program then calculates how the stress wave travels down when the ram strikes the pile head, how much is reflected back up, and how many millimeters the pile penetrates per blow.

The concept comes from the work of E. A. L. Smith in 1960, who proposed dividing the pile into segments and solving the one-dimensional wave equation step by step in time. The software most widely used in pile foundation work today is GRLWEAP, which includes an extensive database of commercial pile hammers.

The key point is that WEA is a prediction before driving, not a measurement. How accurate the results are depends on the quality of the soil data and the assumptions entered.

Why are traditional pile driving formulas not enough?

Because older formulas treat the pile as a rigid body and do not account for stress wave propagation. Dynamic pile driving formulas such as the Engineering News Formula or the Hiley Formula assume that the whole pile moves at once the instant the ram strikes, which is not true for long piles. The stress wave takes on the order of milliseconds to travel from head to toe, so while the pile head is taking the impact, the toe has not yet moved.

As a result, capacities calculated with the older formulas show wide scatter compared with actual test results. Wave Equation Analysis addresses this by simulating wave propagation over time, which explains why 2 piles with the same blow count but driven with different hammer sizes have different capacities.

What data does a wave equation analysis require?

WEA requires 4 groups of data: the hammer, the cushions and helmet, the pile, and the soil model. The accuracy of the analysis depends on the quality of all 4 groups.

1. Hammer

Specify the hammer type (drop hammer, diesel hammer, hydraulic hammer), ram weight, stroke or rated energy per blow, and energy transfer efficiency. Diesel hammers are more complex because their energy also varies with the soil resistance.

2. Cushion & Helmet

The hammer cushion between the hammer and the helmet, and the pile cushion on the head of a concrete pile, blunt the peak of the impact force. Inputs are thickness, area, elastic modulus and coefficient of restitution. Cushion material that deteriorates after driving dozens of piles behaves differently from the values originally assumed.

3. Pile

Inputs are length, cross-sectional area, elastic modulus and unit weight. These values govern 2 different things. The first is the wave speed in the pile material, which depends only on the elastic modulus and density (c = √(E/ρ)). The second is the impedance (Z = EA/c), which governs how the stress wave is reflected and transmitted at splices or changes in cross-section. Prestressed concrete piles and steel piles therefore give clearly different results even with the same design capacity. See how the types differ in Pile types compared

4. Soil Model

The soil model is the most uncertain part. It requires the split of resistance between skin friction and end bearing, the soil quake, and the dynamic damping, based on the soil investigation results and SPT values for the site. Recommended ranges exist from research, but they should always be calibrated against actual test results from nearby projects.

⚠️ Soil data is the core

A wave equation analysis fed with guessed soil data gives results no better than a guess. Use the results of the project's own soil investigation as the basis, not generic textbook values.

The figure below compares the actual driving system with the mass–spring model row by row. Numbers 1–4 are the data groups above, and the right panel is the bearing graph, the output explained in the next section.

Wave Equation Analysis diagram comparing an actual pile driving system with Smith's mass–spring model, and how to read a bearing graph A two-panel diagram with a bottom strip. The left panel places the actual pile driving system on the left and Smith's mass–spring model on the right, with component names in the middle and leader lines to both sides on the same row: the ram, raised through its stroke, is represented by a mass; the hammer cushion by a spring; the helmet by a mass; the pile cushion by a spring; the pile body is divided into mass segments connected by springs; the soil along each segment is represented by a spring with a quake value in parallel with a dashpot with a damping value; and the soil at the pile toe is likewise represented by a spring and dashpot. The actual system shows the pile driven through soft soil into dense sand, with the stress wave traveling down the pile and soil resistance along the shaft and at the toe. The circled numbers one to four are the input data groups. The right panel is a conceptual example bearing graph with no scale: the horizontal axis is blow count, the left vertical axis is ultimate capacity, and the right vertical axis is stress. The capacity curve rises and gradually flattens at high blow counts, the compressive stress curve rises with it, and the tensile stress curve is high at low blow counts and then decreases. Dashed lines show how to read it: start from the required capacity, go across to the curve, then read down to the horizontal axis for the target blow count or driving criterion, then check that the compressive and tensile stresses at the same point do not exceed the allowable values. The bottom strip summarizes the four input groups: the hammer, the cushions and helmet, the pile, and the soil model. Wave Equation Analysis: real system → model → Bearing Graph Not to scale Real driving system ↔ mass–spring model (Smith) Each row is the same part · circled no. = data group Real system Model Soft soil Sand Stroke 1 Ram 2 Hammer pad 2 Helmet 2 Pile cushion 3 Pile (segmented) 4 Shaft soil Quake spring + damping dashpot 4 Toe soil Toe spring + dashpot Stress wave travels down pile Solved per segment, per time step Mass Spring Dashpot (damping) Bearing Graph (output) Conceptual example · no scale Ultimate capacity (Ru) → Pile stress → Ru Comp. Tension Required Ru Target blow count Blow count (blows / set) → A Start at required Ru Go across to the Ru curve B Read down to x-axis Target blow count = driving criterion C Check stress at same point Comp./tension ≤ allowable 1 Hammer Type · ram weight Stroke · efficiency 2 Cushions + helmet Thickness · area E · restitution 3 Pile Length · section E · unit weight 4 Soil model Shaft / toe resistance Quake · Damping
Smith's model represents the ram, helmet and pile segments as masses, the cushions as springs, and the soil as springs (quake) in parallel with dashpots (damping). The output is a bearing graph for reading the target blow count — the graph is a conceptual example, not actual values from any project

How do you read a bearing graph?

The bearing graph is the main output of WEA, showing what capacity each blow count corresponds to. The horizontal axis is the number of blows per unit of penetration (blow count), and the vertical axis shows 3 curves at once: ultimate capacity, maximum compressive stress in the pile, and maximum tensile stress.

Reading it takes 3 steps, as in the right panel of the figure above. First, start from the required capacity and go across to the curve. Second, read down to the horizontal axis for the target blow count, which becomes the driving (set) criterion the site supervisor uses in the field. Third, check that the compressive and tensile stresses at the same blow count do not exceed the allowable values. This criterion is a different thing from the Last 10 Blow count, which measures the actual set after driving is complete.

In the example graph, tensile stress is high at low blow counts, when the pile toe still has little resistance. One graph applies only to that particular combination of hammer, pile and cushion. If the hammer or the cushion thickness changes, the analysis must be rerun.

How does a driveability analysis differ from a bearing graph?

A bearing graph answers "at this blow count, what is the capacity?", while a driveability analysis answers "what blow count will be encountered at each depth?" by entering the soil resistance profile with depth from the soil investigation and letting the program calculate level by level.

The results help assess 3 things that directly affect cost: total blows and driving time per pile, depths at risk of refusal, and depth ranges where pile stresses become abnormally high. A common case on real projects is piles stopping above the design level, explained in the article Piles refusing above the design level

The 3D view below shows what a driveability analysis simulates: the blow count increasing as the pile toe passes through soft clay down into dense sand. The right panel summarizes the sequence for using WEA, from before mobilization to confirmation with PDA.

3D model of a pile driven through soft soil layers into dense sand, as simulated by a wave equation analysis, with the steps for using WEA before mobilization A conceptual 3D view of a soil block with the front corner cut away to show the section. The top layer is soft clay, below it stiff clay, and at the bottom dense sand. A crawler pile driving rig stands on the ground surface with vertical leads; the hammer hangs on a cable above the helmet and cushion on the pile head. The precast concrete pile is shown transparent along its full length, with its toe in the dense sand. An orange wave line shows the compression wave traveling down from head to toe, and a blue wave line shows the wave reflected back from the toe. Green arrows show resistance along the shaft and at the toe. On the section face beside the pile, horizontal bars show blow count versus depth as a conceptual example: short bars in the soft clay, longer in the stiff clay and longest in the dense sand, which is what a driveability analysis evaluates. Labels on the left explain that soft clay gives a low blow count but a risk of tensile stress while the toe has no resistance, whereas dense sand gives a high blow count, high compressive stress at the toe, and is the zone at risk of refusal. The right panel lists five steps for using WEA before mobilization: collect the four data groups including actual soil investigation results; run the model for each hammer option; read the bearing graph and driveability results; select the hammer, stroke and cushion and set the driving criterion; then confirm in the field with PDA and CAPWAP on representative piles to adjust the soil values in the model and issue a revised criterion. A note states that WEA gives predicted values and that capacity must be confirmed by field testing. 3D model: driving a pile through soil layers + steps for using WEA Not to scale Drop hammer Raised and dropped by stroke Helmet + cushion Blunts the impact peak Comp. wave down pile Reaches toe in milliseconds Soft clay Low blow count High stroke → tension risk Stiff clay Resistance increases Sand High blow count · high toe stress Zone at risk of refusal Pile driving rig Leads guide the hammer Blow count vs depth Driveability · conceptual example Wave reflected from toe Shaft / toe resistance WEA steps before mobilization 1 Collect 4 data groups Hammer · cushion · pile + actual soil boring results 2 Run the model e.g. GRLWEAP per hammer the contractor actually has 3 Read results Bearing Graph · Driveability Comp. / tensile stress 4 Decide before driving Pick hammer · stroke · cushion Set driving criterion 5 Confirm in field with PDA CAPWAP on test piles Adjust soil → new criterion WEA = prediction before driving Confirm actual capacity with field tests such as PDA per DPT 1252-51 · ASTM D4945
The orange bars on the section face show blow count versus depth as a conceptual example. The bars lengthen in the dense sand layer, the zone at risk of refusal, while in the soft soil zone tensile stress must be watched. Analysis results are predictions that must be confirmed by field testing — conceptual image, not the layout or values of any actual project

Driving stress: why do piles break even when the hammer is not large?

Piles can break during driving even with a modest hammer, because damage depends on the peak stress from the stress wave, not on hammer weight alone. During driving, the pile material experiences 2 kinds of stress alternating within a fraction of a second.

  • Compressive stress — peaks at the pile head from the direct impact and can spall the head. When the toe hits dense sand or rock, the compression wave reflected from the toe superimposes on the downward wave, so the compressive stress at the toe may exceed that at the head. Both locations must be checked.
  • Tensile stress — occurs when the compression wave reaches a toe that has no resistance yet and is reflected back as tension. It is most common while the pile is still in soft soil but driven with a high stroke, and it causes transverse cracks along the pile in prestressed concrete piles (see other types of damage in Pile defects: causes and detection)

Common remedies are increasing the pile cushion thickness to blunt the peak force, or reducing the hammer stroke while the pile is still in soft soil and increasing it once resistance begins to build. Allowable driving stresses for each material must follow the project specifications and the design engineer.

How heavy a hammer is needed to drive the pile? How WEA helps

There is no ready-made answer to this question, because it depends on the soil layers, the pile size and length, and the equipment actually available on site. But the selection principle you can decide on is finding the balance between two risks.

← Scroll to see the full table →

CaseWhat happensConsequence
Hammer too smallEnergy per blow is not enough to overcome the resistanceBlow count shoots up and the pile stops before reaching the design level — false refusal, lost time and a risk of cumulative pile head damage
Hammer too largePeak impact force exceeds the material strengthCompressive stress at the head or tensile stress along the pile exceeds the limit; the pile cracks or breaks during driving
Hammer just rightBlow count stays in a controllable range and stresses are below the limitsThe pile reaches the design level without damage and the schedule stays under control

The way to find the right fit is to run a wave equation analysis for each hammer the contractor actually has, then compare both the blow counts and the stresses obtained, rather than choosing on experience alone.

How does WEA work together with PDA and static load tests?

WEA is used for planning before driving, while PDA and static load tests measure actual values on piles that have been driven. All 3 belong to different stages of the work and cannot replace one another.

← Scroll to see the full table →

TopicWave Equation AnalysisDynamic Load Test (PDA)Static Load Test
TimingBefore drivingDuring or after drivingAfter the pile has aged
Type of dataPredicted values from a modelMeasured values from sensors on the pileMeasured values from applied load
AnswersSelect the hammer, set the driving criterion, assess the risk of pile damageCapacity, integrity, transferred energyCapacity and settlement under actual load
Reference standardsNo dedicated test standard; used as an analysis toolDPT 1252-51 · ASTM D4945ASTM D1143 · DPT 1251
Can it confirm capacity per the specification?NoYesYes

In practice, all 3 are linked in a loop (step 5 in the 3D view above): run WEA to select the hammer and set the criterion before driving, then perform a Dynamic Load Test (PDA) on representative piles, then feed the CAPWAP analysis results back to adjust the soil values in the model to match the actual site, and issue a revised driving criterion for the remaining piles. This gives the field criterion a basis in measured values rather than assumptions alone.

For piles where soil strength gain over time needs to be assessed, also read Restrike test and soil set-up

How much do WEA and related tests cost?

Wave Equation Analysis has no fixed unit price, because fees depend on the project scope, the number of hammer options and the pile types to compare. SPN does not take on WEA analysis work directly, but provides the field tests that usually follow, with starting prices as follows:

  • Dynamic Load Test (PDA) from about THB 5,500/pile
  • Static Load Test about THB 50,000–150,000+/pile, depending on the test load and the reaction system

These prices exclude travel and equipment mobilization, which are charged extra according to the distance to site. See more details in Pile testing prices 2026 or contact us for a free quote within 24 hours.

What are the limitations of wave equation analysis?

WEA has 4 main limitations to know before using the results.

  • Results are only as accurate as the input — soil resistance, quake and damping are estimates; if the actual soil layers differ from the assumptions, the results deviate accordingly.
  • Actual hammer efficiency differs from tabulated values — hammers in heavy use or poorly maintained transfer less energy than the specified values.
  • It does not cover pile integrity — WEA indicates risk from stresses but does not check whether a pile is already cracked. That requires testing such as the Seismic Integrity Test per DPT 1551-51.
  • It does not replace specified testing — confirming capacity for submission to the relevant authorities still requires field test results certified by a licensed civil engineer registered with the Council of Engineers Thailand (COE).

Summary

Wave Equation Analysis is a planning tool used before pile driving that turns the question "how heavy a hammer will drive the pile?" from guesswork into numbers that can be checked. The bearing graph is used to set the driving criterion, and the driveability analysis is used to assess the risk of refusal and stresses with depth. Both reduce the chance of pile damage and delays, but all results are predictions. Confirming actual capacity still requires field testing, such as PDA per DPT 1252-51 and ASTM D4945, or a static load test per ASTM D1143 and DPT 1251.

Note that SPN Soil Engineering does not provide wave equation analysis (GRLWEAP) directly — this article is written for educational purposes — but we do provide the field tests used to confirm actual performance: Dynamic Load Test (PDA), Static Load Test and Seismic Integrity Test (see the overview at Pile testing services). If you are planning pile driving work, SPN's engineering team is happy to advise and plan testing that meets the project specifications.